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Hessian equations of Krylov type on compact Hermitian manifolds

Hessian equations of Krylov type on compact Hermitian manifolds 1IntroductionLet (M,ω)\left(M,\omega )be a compact Kähler manifold of complex dimension nn. In 1978, Yau [1] proved the famous Calabi-Yau conjecture by solving the following complex Monge-Ampère equation on MM(ω+−1∂∂¯u)n=fωn,{\left(\omega +\sqrt{-1}\partial \overline{\partial }u)}^{n}=f{\omega }^{n},with positive function ff. There have been many generalizations of Yau’s work. One extension of Yau’s Theorem is to the case of Hermitian manifolds, which is initiated by Cherrier [2] in 1987. The Monge-Ampère equation on compact Hermitian manifolds was solved by Tosatti and Weinkove [3], building on several earlier works. See [2,4,5, 6,7] and the references therein.The complex Hessian equation can be expressed as follows: (ω+−1∂∂¯u)k∧ωn−k=fωn,2≤k≤n−1.{\left(\omega +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=f{\omega }^{n},\hspace{1.0em}2\le k\le n-1.On compact Kähler manifolds (M,ω)\left(M,\omega ), Hou [8] proved the existence of a smooth admissible solution of the complex Hessian equation by assuming the nonnegativity of the holomorphic bisectional curvature. Later, Hou et al. [9] obtained the second-order estimate without any curvature assumption. Using Hou et al.’s estimate, Dinew and Kolodziej [10] applied a blow-up argument to prove the gradient estimate and solved the complex Hessian equation on compact Kähler manifolds. The corresponding problem on Hermitian manifolds was solved by Zhang [11] and Székelyhidi [12] independently.The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let χ\chi be a real (1, 1) form, the complex Hessian quotient equations can be expressed as follows: (χ+−1∂∂¯u)k∧ωn−k=f(z)(χ+−1∂∂¯u)l∧ωn−l,1≤l<k≤n,z∈M.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=f\left(z){\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{l}\wedge {\omega }^{n-l},\hspace{1.0em}1\le l\lt k\le n,z\in M.When f(z)f\left(z)is constant, one special case is the so-called Donaldson equation [13]: (χ+−1∂∂¯u)n=c(χ+−1∂∂¯u)n−1∧ωk.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}=c{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-1}\wedge {\omega }^{k}.After some progresses made in [14,15, 16,17], Song and Weinokove [18] solved the Donaldson equation on closed Kähler manifolds via JJ-flow. Fang et al. [19] extended the Donaldson equation to (χ+−1∂∂¯u)n=ck(χ+−1∂∂¯u)n−k∧ωk{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}={c}_{k}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-k}\wedge {\omega }^{k}and solved this equation on closed Kähler manifolds by assuming a cone condition. When f(z)f\left(z)is not constant, analogous results were obtained by Sun [20,21] on compact Hermitian manifolds.In this article, we are concerned with Hessian equations of Krylov type in the form of the linear combinations of the Hessian, which can be written as follows: (1.1)(χ+−1∂∂¯u)k∧ωn−k=∑l=0k−1αl(χ+−1∂∂¯u)l∧ωn−l,2≤k≤n.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{l}\wedge {\omega }^{n-l},\hspace{1.0em}2\le k\le n.The Dirichlet problem of (1.1) on (k−1)\left(k-1)-convex domain Ω\Omega in Rn{{\mathbb{R}}}^{n}was first studied by Krylov [22] about 20 years ago. He observed that if αl(x)≥0{\alpha }_{l}\left(x)\ge 0for 0≤l≤k−10\le l\le k-1, the natural admissible cone to make (1.1) elliptic is also the Γk{\Gamma }_{k}-cone, which is the same as the kk-Hessian equation case, where Γk={λ∈Rn∣σ1(λ)>0,…,σk(λ)>0}.{\Gamma }_{k}=\left\{\lambda \in {{\mathbb{R}}}^{n}| {\sigma }_{1}\left(\lambda )\gt 0,\ldots ,{\sigma }_{k}\left(\lambda )\gt 0\right\}.Guan and Zhang [23] solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali [24] proved a priori estimates to the following equation in Kähler case: (χ+−1∂∂¯u)n=∑l=0n−1Cnlαl(χ+−1∂∂¯u)n−k∧ωl,{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}=\mathop{\sum }\limits_{l=0}^{n-1}{C}_{n}^{l}{\alpha }_{l}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-k}\wedge {\omega }^{l},where αl≥0{\alpha }_{l}\ge 0are smooth real functions such that either αl=0{\alpha }_{l}=0or αl>0{\alpha }_{l}\gt 0, and ∑l=0n−2αl>0\mathop{\sum }\limits_{l=0}^{n-2}{\alpha }_{l}\gt 0. Recently, Phong and Tô [25] solved Hessian equations of Krylov type on compact Kähler manifolds, where αl{\alpha }_{l}are non-negative constants for 0≤l≤k−10\le l\le k-1. When αl{\alpha }_{l}are non-negative smooth functions for 0≤l≤k−10\le l\le k-1, analogous results on compact Kähler manifolds are obtained by Chen [26] and Zhou [27] independently.Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou [27] believed that the condition on αk−1(x)>0{\alpha }_{k-1}\left(x)\gt 0is not necessary. In fact, Guan and Zhang [23] considered equation (1.1) without the sign requirement for the coefficient function αk−1(x){\alpha }_{k-1}\left(x).In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for αk−1(x){\alpha }_{k-1}\left(x). Let Γk−1g{\Gamma }_{k-1}^{g}be the set of all the real (1, 1) forms, eigenvalues of which belong to Γk−1{\Gamma }_{k-1}. To ensure the ellipticity and non degeneracy of the equation in Γk−1{\Gamma }_{k-1}, we require smooth real functions αl{\alpha }_{l}to satisfy the conditions: for 0≤l≤k−20\le l\le k-2, either αl>0{\alpha }_{l}\gt 0or αl≡0{\alpha }_{l}\equiv 0, and ∑l=0k−2αl>0{\sum }_{l=0}^{k-2}{\alpha }_{l}\gt 0. Let χu=χ+−1∂∂¯u{\chi }_{u}=\chi +\sqrt{-1}\partial \bar{\partial }uand χu̲=χ+−1∂∂¯u̲{\chi }_{\underline{u}}=\chi +\sqrt{-1}\partial \bar{\partial }\underline{u}. To state our main results, we need also the following condition of C{\mathcal{C}}-subsolution, which is similar to C{\mathcal{C}}-subsolution introduced by Székelyhidi [12].Definition 1.1A smooth real function u̲\underline{u}is a C{\mathcal{C}}-subsolution to (1.1), if χu̲∈Γk−1g,{\chi }_{\underline{u}}\in {\Gamma }_{k-1}^{g},and at each point x∈Mx\in M, the set λ(χ˜)∈Γk−1∣χ˜k∧ωn−k=∑l=0k−1αl(x)χ˜l∧ωn−landχ˜−χu̲≥0\left\{\lambda \left(\widetilde{\chi })\in {\Gamma }_{k-1}| {\widetilde{\chi }}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}\left(x){\widetilde{\chi }}^{l}\wedge {\omega }^{n-l}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{\chi }-{\chi }_{\underline{u}}\ge 0\right\}is bounded.Theorem 1.2Let (M,g)\left(M,g)be a compact Hermitian manifold, χ\chi a real (1, 1) form on MM. Suppose that u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1) and at each point x∈Mx\in M, (1.2)χu̲(x)k∧ωn−k≤∑l=0k−1αl(x)χu̲(x)l∧ωn−l.{\chi }_{\underline{u}\left(x)}^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}\left(x){\chi }_{\underline{u}\left(x)}^{l}\wedge {\omega }^{n-l}.Then there exists a smooth real function uuon MMand a unique constant bbsolving(1.3)χuk∧ωn−k=∑l=0k−2αl(x)χul∧ωn−l+(αk−1+b)χuk−1∧ωn−k+1,{\chi }_{u}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\chi }_{u}^{l}\wedge {\omega }^{n-l}+\left({\alpha }_{k-1}+b){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1},with supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0and χu∈Γk−1g{\chi }_{u}\in {\Gamma }_{k-1}^{g}.Corollary 1.3Let (M,g)\left(M,g)be a compact Kähler manifold, χ\chi a closed (1,1)\left(1,\hspace{0.33em}1)-form. Suppose that u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1) and(1.4)∫Mχk∧ωn−k≤∑l=0k−1cl∫Mχ∧ωn−l,\mathop{\int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-1}{c}_{l}\mathop{\int }\limits_{M}\chi \wedge {\omega }^{n-l},where cl=infMαl{c}_{l}={\inf }_{M}{\alpha }_{l}, 0≤l≤k−10\le l\le k-1. Then there exists a smooth real function u on M and a unique constant b solving(1.5)χuk∧ωn−k=∑l=0k−2αl(x)χul∧ωn−l+(αk−1+b)χuk−1∧ωn−k+1,{\chi }_{u}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\chi }_{u}^{l}\wedge {\omega }^{n-l}+\left({\alpha }_{k-1}+b){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1},with supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0and χu∈Γk−1g{\chi }_{u}\in {\Gamma }_{k-1}^{g}.Lately, Pingali [28] proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle θˆ∈12π,32π\hat{\theta }\in \left(\frac{1}{2}\pi ,\frac{3}{2}\pi \right)on compact Kähler threefold. Let Ω\Omega be a closed (1,1)\left(1,\hspace{0.33em}1)form, Ωu=Ω+−1∂∂¯u{\Omega }_{u}=\Omega +\sqrt{-1}\partial \bar{\partial }u. From [24], the deformed Hermitian Yang-Mills equation on compact Kähler threefold can be written as follows: (1.6)Ωu3=3sec2(θˆ)Ωu∧ω2+2tan(θˆ)sec2(θˆ)ω3.{\Omega }_{u}^{3}=3{{\rm{\sec }}}^{2}\left(\hat{\theta }){\Omega }_{u}\wedge {\omega }^{2}+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta }){\omega }^{3}.As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.Corollary 1.4Let (M,g)\left(M,g)be a compact Kähler threefold, constant phase angle θˆ∈12π,32π\hat{\theta }\in \left(\frac{1}{2}\pi ,\frac{3}{2}\pi \right), and Ω\Omega a positive definite closed (1,1)\left(1,\hspace{0.33em}1)form, satisfying the following conditions: (1.7)3Ω2−3sec2(θˆ)ω2>0,3{\Omega }^{2}-3{{\rm{\sec }}}^{2}\left(\hat{\theta }){\omega }^{2}\gt 0,(1.8)∫MΩ3=3sec2(θˆ)∫MΩ∧ω2+2tan(θˆ)sec2(θˆ)∫Mω3,\mathop{\int }\limits_{M}{\Omega }^{3}=3{{\rm{\sec }}}^{2}\left(\hat{\theta })\mathop{\int }\limits_{M}\Omega \wedge {\omega }^{2}+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\mathop{\int }\limits_{M}{\omega }^{3},(1.9)Ω+sec(θˆ)ω∈Γ2g.\Omega +{\rm{\sec }}\left(\hat{\theta })\omega \in {\Gamma }_{2}^{g}.Then there exists a smooth solution to equation (1.6) with supMu=0{\sup }_{M}u=0and Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.The rest of this article is organized as follows. In Section 2, we set up some notations and provide some preliminary results. In Section 3, we give the C0{C}^{0}estimate by the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, we establish the C2{C}^{2}estimate for equation (1.1) by the method of Hou et al. [9] and the C{\mathcal{C}}-subsolution condition. In Section 5, we give the gradient estimate. In Section 6, we give the proof of Theorem 1.2, Corollaries 1.3, and 1.4 by the method of continuity. Although the method is very standard in the study of elliptic PDEs, it is not easy to carry out this method on a compact Hermitian manifold. Since the essential C{\mathcal{C}}-subsolution condition depends on α0,…,αk−1{\alpha }_{0},\ldots ,{\alpha }_{k-1}, we have to find a uniform C{\mathcal{C}}-subsolution condition for the solution flow of the continuity method.2PreliminariesIn this section, we set up the notation and establish some lemmas. Let σk(λ){\sigma }_{k}\left(\lambda )denote the kkth elementary symmetric function σk(λ)=∑1≤i1<⋯<ik≤nλi1⋯λik,forλ=(λ1,…,λn)∈Rn,1≤k≤n.{\sigma }_{k}\left(\lambda )=\sum _{1\le {i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{k}\le n}{\lambda }_{{i}_{1}}\cdots {\lambda }_{{i}_{k}},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{1.0em}\lambda =\left({\lambda }_{1},\ldots ,{\lambda }_{n})\in {{\mathbb{R}}}^{n},\hspace{1.0em}1\le k\le n.For completeness, we define σ0(λ)=1{\sigma }_{0}\left(\lambda )=1and σ−1(λ)=0{\sigma }_{-1}\left(\lambda )=0. Let σk(λ∣i){\sigma }_{k}\left(\lambda | i)denote the symmetric function with λi=0{\lambda }_{i}=0and σk(λ∣ij){\sigma }_{k}\left(\lambda | ij)the symmetric function with λi=λj=0{\lambda }_{i}={\lambda }_{j}=0. Also denote by σk(A∣i){\sigma }_{k}\left(A| i)the symmetric function with AAdeleting the iith row and iith column, and σk(A∣ij){\sigma }_{k}\left(A| ij)the symmetric function with AAdeleting the iith, jjth rows and iith, jjth columns, for all 1≤i,j≤n1\le i,j\le n. In local coordinates, Xij¯=X∂∂zi,∂∂z¯i=χij¯+uij¯,X̲ij¯=χij¯+u̲ij¯.{X}_{i\overline{j}}=X\left(\frac{\partial }{\partial {z}^{i}},\frac{\partial }{\partial {\overline{z}}^{i}}\right)={\chi }_{i\overline{j}}+{u}_{i\overline{j}},\hspace{1em}{\underline{X}}_{i\overline{j}}={\chi }_{i\overline{j}}+{\underline{u}}_{i\overline{j}}.Define λ(χu)\lambda \left({\chi }_{u})as the eigenvalue set of {Xij¯}\left\{{X}_{i\overline{j}}\right\}with respect to {gij¯}\left\{{g}_{i\overline{j}}\right\}. In local coordinates, equation (1.1) can be written in the following form: (2.1)σk(λ(χu))=∑l=0k−1βl(x)σl(λ(χu)),{\sigma }_{k}\left(\lambda \left({\chi }_{u}))=\mathop{\sum }\limits_{l=0}^{k-1}{\beta }_{l}\left(x){\sigma }_{l}\left(\lambda \left({\chi }_{u})),where σl(λ(χu))Cnl=χul∧ωn−lωn,βl(x)=CnkCnlαl(x).\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{C}_{n}^{l}}=\frac{{\chi }_{u}^{l}\wedge {\omega }^{n-l}}{{\omega }^{n}},\hspace{1em}{\beta }_{l}\left(x)=\frac{{C}_{n}^{k}}{{C}_{n}^{l}}{\alpha }_{l}\left(x).Equivalently, we can rewrite equation (2.1) as follows: (2.2)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=βk−1(x).\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}={\beta }_{k-1}\left(x).Lemma 2.1[29,30] For λ∈Γm\lambda \in {\Gamma }_{m}and m>l≥0m\gt l\ge 0, r>s≥0r\gt s\ge 0, m≥rm\ge r, l≥sl\ge s, we haveσm(λ)/Cnmσl(λ)/Cnl1m−l≤σr(λ)/Cnrσs(λ)/Cns1r−s.{\left[\frac{{\sigma }_{m}\left(\lambda )\text{/}{C}_{n}^{m}}{{\sigma }_{l}\left(\lambda )\text{/}{C}_{n}^{l}}\right]}^{\tfrac{1}{m-l}}\le {\left[\frac{{\sigma }_{r}\left(\lambda )\text{/}{C}_{n}^{r}}{{\sigma }_{s}\left(\lambda )\text{/}{C}_{n}^{s}}\right]}^{\tfrac{1}{r-s}}.The following lemma is similar to Lemma 2.3 in [27], but we need to discuss it more widely, that is, λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}instead of λ(χu)∈Γk\lambda \left({\chi }_{u})\in {\Gamma }_{k}.Lemma 2.2If u∈C2(M)u\in {C}^{2}\left(M)is a solution of (2.2), λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and βl(x)>0{\beta }_{l}\left(x)\gt 0, 0≤l≤k−20\le l\le k-2, then(2.3)σl(λ)σk−1(λ)≤C(n,k,inf0≤l≤k−2βl,sup∣βk−1∣)for0≤l≤k−2.\frac{{\sigma }_{l}\left(\lambda )}{{\sigma }_{k-1}\left(\lambda )}\le C\left(n,k,\mathop{\inf }\limits_{0\le l\le k-2}{\beta }_{l},\sup | {\beta }_{k-1}| \right)\hspace{1.0em}{for}\hspace{0.33em}0\le l\le k-2.ProofIf σkσk−1≤1\frac{{\sigma }_{k}}{{\sigma }_{k-1}}\le 1, then we obtain from equation (2.2) βl(x)σlσk−1≤σkσk−1−β(x)≤1−β(x)≤C(supM∣βk−1∣),for0≤l≤k−2.{\beta }_{l}\left(x)\frac{{\sigma }_{l}}{{\sigma }_{k-1}}\le \frac{{\sigma }_{k}}{{\sigma }_{k-1}}-\beta \left(x)\le 1-\beta \left(x)\le C\left(\mathop{\sup }\limits_{M}| {\beta }_{k-1}| \right),\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}0\le l\le k-2.If σkσk−1>1\frac{{\sigma }_{k}}{{\sigma }_{k-1}}\gt 1, i.e., σk−1σk<1\frac{{\sigma }_{k-1}}{{\sigma }_{k}}\lt 1, we see from Lemma 2.1 that σlσk−1≤(Cnk)k−1−lCnl(Cnk−1)k−lσk−1σkk−1−l≤(Cnk)k−1−lCnl(Cnk−1)k−l≤C(n,k)\frac{{\sigma }_{l}}{{\sigma }_{k-1}}\le \frac{{\left({C}_{n}^{k})}^{k-1-l}{C}_{n}^{l}}{{\left({C}_{n}^{k-1})}^{k-l}}{\left(\frac{{\sigma }_{k-1}}{{\sigma }_{k}}\right)}^{k-1-l}\le \frac{{\left({C}_{n}^{k})}^{k-1-l}{C}_{n}^{l}}{{\left({C}_{n}^{k-1})}^{k-l}}\le C\left(n,k)for 0≤l≤k−20\le l\le k-2, which completes the proof of Lemma 2.2.□For any point x0∈M{x}_{0}\in M, choose a local frame such that Xij¯=δijXii¯{X}_{i\overline{j}}={\delta }_{ij}{X}_{i\overline{i}}. For the convenience of notations, we will write equation (2.2) as follows: (2.4)F(X)=Fk(X)+∑l=0k−2βlFl(X)=βk−1(x),F\left(X)={F}_{k}\left(X)+\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}{F}_{l}\left(X)={\beta }_{k-1}\left(x),where Fk(X)=σk(λ(X))σk−1(λ(X)){F}_{k}\left(X)=\frac{{\sigma }_{k}\left(\lambda \left(X))}{{\sigma }_{k-1}\left(\lambda \left(X))}and Fl(X)=−σl(λ(X))σk−1(λ(X)){F}_{l}\left(X)=-\frac{{\sigma }_{l}\left(\lambda \left(X))}{{\sigma }_{k-1}\left(\lambda \left(X))}. Let Fij¯≔∂F∂Xij¯=∂F∂λk∂λk∂Xij¯,{F}^{i\overline{j}}:= \frac{\partial F}{\partial {X}_{i\overline{j}}}=\frac{\partial F}{\partial {\lambda }_{k}}\frac{\partial {\lambda }_{k}}{\partial {X}_{i\overline{j}}},then at x0{x}_{0}, we have Fij¯=Fii¯δij.{F}^{i\overline{j}}={F}^{i\overline{i}}{\delta }_{ij}.Let ℱ≔∑iFii¯.{\mathcal{ {\mathcal F} }}:= \sum _{i}{F}^{i\overline{i}}.Lemma 2.3[23] If λ∈Γk−1\lambda \in {\Gamma }_{k-1}and αl(x)>0{\alpha }_{l}\left(x)\gt 0, 0≤l≤k−20\le l\le k-2, then the operator F is elliptic and concave in Γk−1{\Gamma }_{k-1}.From Lemma 2.4 in [27] and Lemma 2.2, we have the following lemma.Lemma 2.4If u∈C2(M)u\in {C}^{2}\left(M)is a solution of (2.4), λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, then at x0{x}_{0}, (2.5)n−k+1k≤ℱ≤C(n,k,inf0≤l≤k−2αl,sup∣αk−1∣).\frac{n-k+1}{k}\le {\mathcal{ {\mathcal F} }}\le C\left(n,k,\mathop{\inf }\limits_{0\le l\le k-2}{\alpha }_{l},\sup | {\alpha }_{k-1}| \right).Lemma 2.5Under assumptions of Theorem 1.2, there is a constant θ>0\theta \gt 0such that(2.6)Fii¯(uii¯−u̲ii¯)≤−θ(1+ℱ),{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})\le -\theta \left(1+{\mathcal{ {\mathcal F} }}),or(2.7)F11¯≥θ.{F}^{1\overline{1}}\ge \theta .ProofWithout loss of generality, we may assume that X11¯≥⋯≥Xnn¯{X}_{1\overline{1}}\hspace{0.33em}\ge \cdots \ge {X}_{n\overline{n}}. Thus, Fnn¯≥⋯≥F11¯.{F}^{n\overline{n}}\hspace{0.33em}\ge \cdots \ge {F}^{1\overline{1}}.Since u̲\underline{u}is a C{\mathcal{C}}-subsolution, if ε>0\varepsilon \gt 0is small enough, χu̲−εω{\chi }_{\underline{u}}-\varepsilon \omega still satisfies Definition 1.1. Since MMis compact, there are uniform constants N>0N\gt 0and δ>0\delta \gt 0such that (2.8)F(X˜)>βk−1+δ,F\left(\widetilde{X})\gt {\beta }_{k-1}+\delta ,where X˜=X̲−εg+N0⋯000⋯000⋯0n×n.\widetilde{X}=\underline{X}-\varepsilon g+{\left(\begin{array}{cccc}N& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ 0& 0& \cdots & 0\end{array}\right)}_{n\times n}.Direct calculation yields (2.9)Fii¯(uii¯−u̲ii¯)=Fii¯(Xii¯−X̲ii¯)=Fii¯(Xii¯−X˜ii¯)+F11¯N−εℱ.{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})={F}^{i\overline{i}}\left({X}_{i\overline{i}}-{\underline{X}}_{i\overline{i}})={F}^{i\overline{i}}({X}_{i\overline{i}}-{\widetilde{X}}_{i\overline{i}})+{F}^{1\overline{1}}N-\varepsilon {\mathcal{ {\mathcal F} }}.Since FFis concave in Γk−1{\Gamma }_{k-1}, from (2.8), we obtain (2.10)∑i=1nFii¯(Xii¯−X˜ii¯)≤F(X)−F(X˜)≤−δ.\mathop{\sum }\limits_{i=1}^{n}{F}^{i\overline{i}}\left({X}_{i\overline{i}}-{\widetilde{X}}_{i\overline{i}})\le F\left(X)-F\left(\widetilde{X})\le -\delta .By substituting (2.10) into (2.9), we obtain Fii¯(uii¯−u̲ii¯)≤−δ+F11¯N−εℱ.{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})\le -\delta +{F}^{1\overline{1}}N-\varepsilon {\mathcal{ {\mathcal F} }}.Set θ=minδ2,ε,δ2N\theta =\min \left\{\frac{\delta }{2},\varepsilon ,\frac{\delta }{2N}\right\}. If F11¯N≤δ2{F}^{1\overline{1}}N\le \frac{\delta }{2}, we have (2.6); otherwise, (2.7) must be true.□3C0{C}^{0}estimateIn this section, we obtain the C0{C}^{0}estimate by using the Alexandroff-Bakelman-Pucci maximum principle and prove the following Proposition 3.1, which is similar to the approach of Székelyhidi [12].Proposition 3.1Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. We normalize u such that supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0. There is a constant C depending on the given data, such that(3.1)supM∣u∣<C.\mathop{\sup }\limits_{M}| u| \lt C.ProofTo simplify notation, we can assume u̲=0\underline{u}=0, otherwise we modify the background form χ\chi . Therefore, supMu=0{\sup }_{M}u=0. The following goal is to prove that L=infMuL={\inf }_{M}uhas a uniform lower bound. Notice that λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, so λ(χu)∈Γ1\lambda \left({\chi }_{u})\in {\Gamma }_{1}, that is, Δu=gip¯uip¯>−gip¯χip¯≥−C^.\Delta u={g}^{i\overline{p}}{u}_{i\overline{p}}\gt -{g}^{i\overline{p}}{\chi }_{i\overline{p}}\ge -\widehat{C}.Let G:M×M→RG:M\times M\to {\mathbb{R}}be the Green’s function of a Gauduchon metric conformal to gg. From [1], there is a uniform constant KKsuch that G(x,y)+K≥0,∀(x,y)∈M×M,and∫y∈MG(x,y)ωn(y)=0.G\left(x,y)+K\ge 0,\hspace{1em}\forall \left(x,y)\in M\times M,\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\mathop{\int }\limits_{y\in M}G\left(x,y){\omega }^{n}(y)=0.Since supMu=0{\sup }_{M}u=0, there is a point x0∈M{x}_{0}\in Msuch that u(x0)=0u\left({x}_{0})=0. Hence, u(x0)=∫Mudμ−∫y∈MG(x0,y)Δu(y)ωn(y)=∫Mudμ−∫y∈M(G(x0,y)+K)Δu(y)ωn(y)≤∫Mudμ+C^K,\begin{array}{rcl}u\left({x}_{0})& =& \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu -\mathop{\displaystyle \int }\limits_{y\in M}G\left({x}_{0},y)\Delta u(y){\omega }^{n}(y)\\ & =& \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu -\mathop{\displaystyle \int }\limits_{y\in M}\left(G\left({x}_{0},y)+K)\Delta u(y){\omega }^{n}(y)\\ & \le & \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu +\widehat{C}K,\end{array}which yields ∫M∣u∣dμ≤C^K.\mathop{\int }\limits_{M}| u| {\rm{d}}\mu \le \widehat{C}K.Next, we choose local coordinates at the minimum point of uuand L=infMu=u(0)L={\inf }_{M}u=u\left(0). Let B(1)={z:∣z∣<1}B\left(1)=\left\{z:| z| \lt 1\right\}and v=u+ε∣z∣2v=u+\varepsilon | z\hspace{-0.25em}{| }^{2}for a small ε>0\varepsilon \gt 0. From the Alexandroff-Bakelman-Pucci maximum principle, we obtain (3.2)c0ε2n≤∫Ωdet(D2v),{c}_{0}{\varepsilon }^{2n}\le \mathop{\int }\limits_{\Omega }\det \left({D}^{2}v),where Ω=x∈B(1):∣Dv(x)∣<ε2,v(y)≥v(x)+Dv(x)⋅(y−x),∀y∈B(1).\Omega =\left\{\begin{array}{c}x\in B\left(1):| Dv\left(x)| \lt \frac{\varepsilon }{2},\\ v(y)\ge v\left(x)+Dv\left(x)\cdot (y-x),\hspace{1em}\forall y\in B\left(1)\end{array}\right\}.Let λ˜−λ(χu̲)≥0\widetilde{\lambda }-\lambda \left({\chi }_{\underline{u}})\ge 0and σk(λ˜)σk−1(λ˜)−∑l=0k−2βlσl(λ˜)σk−1(λ˜)=βk−1(x).\frac{{\sigma }_{k}\left(\widetilde{\lambda })}{{\sigma }_{k-1}\left(\widetilde{\lambda })}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\widetilde{\lambda })}{{\sigma }_{k-1}\left(\widetilde{\lambda })}={\beta }_{k-1}\left(x).u̲\underline{u}is a C{\mathcal{C}}subsolution, which means that ∣λ˜∣| \widetilde{\lambda }| is bounded. Since MMis compact, there is uniform constant η>0\eta \gt 0such that λ(χu̲)−η1\lambda \left({\chi }_{\underline{u}})-\eta {\bf{1}}satisfies Definition 1.1. Since Ω\Omega is a contact set, we have D2v(x)≥0{D}^{2}v\left(x)\ge 0, for x∈Ωx\in \Omega , which implies uij¯(x)+εδij≥0{u}_{i\overline{j}}\left(x)+\varepsilon {\delta }_{ij}\ge 0. Choosing ε\varepsilon such that 0<ε≤η0\lt \varepsilon \le \eta , on Ω\Omega , we have λ(χu)−(λ(χu̲)−η1)≥λ(χu)−(λ(χu̲)−ε1)=λ(uij¯)+ε1≥0.\lambda \left({\chi }_{u})-(\lambda \left({\chi }_{\underline{u}})-\eta {\bf{1}})\ge \lambda \left({\chi }_{u})-(\lambda \left({\chi }_{\underline{u}})-\varepsilon {\bf{1}})=\lambda \left({u}_{i\overline{j}})+\varepsilon {\bf{1}}\ge 0.Since σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=βk−1(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}={\beta }_{k-1}\left(x),we obtain that ∣λ(χu)∣| \lambda \left({\chi }_{u})| is bounded, which yields ∣uij¯∣≤C| {u}_{i\overline{j}}| \le C. Then det(D2v(x))≤22ndet(vij¯)2≤C.\det \left({D}^{2}v\left(x))\le {2}^{2n}\det {\left({v}_{i\overline{j}})}^{2}\le C.From this and (3.2), we obtain (3.3)c0ε2n≤∫Ωdet(D2v)≤C⋅vol(Ω).{c}_{0}{\varepsilon }^{2n}\le \mathop{\int }\limits_{\Omega }{\rm{\det }}\left({D}^{2}v)\le C\cdot {\rm{vol}}\left(\Omega ).On the other hand, we have for x∈Ωx\in \Omega v(0)≥v(x)−Dv(x)⋅x>v(x)−ε2,v\left(0)\ge v\left(x)-Dv\left(x)\cdot x\gt v\left(x)-\frac{\varepsilon }{2},so ∣v(x)∣>∣L+ε2∣.| v\left(x)| \gt | L+\frac{\varepsilon }{2}| .Then, ∫M∣v(x)∣≥∫Ω∣v(x)∣≥∣L+ε2∣⋅vol(Ω).\mathop{\int }\limits_{M}| v\left(x)| \ge \mathop{\int }\limits_{\Omega }| v\left(x)| \ge | L+\frac{\varepsilon }{2}| \cdot {\rm{vol}}\left(\Omega ).Since ∫M∣v(x)∣\mathop{\int }\limits_{M}| v\left(x)| is uniformly bounded, this inequality contradicts (3.3) if LLis very large.□4C2{C}^{2}estimateIn this section, we establish the C2{C}^{2}estimate to equation (1.1). Our calculation is similar to that in [27], but on Hermitian manifolds, equation (1.1) are much more difficult to treat due to the torsion terms.4.1Notations and lemmaIn local coordinates z=(z1,…,zn)z=\left({z}_{1},\ldots ,{z}_{n}), the Chern connection ∇\nabla and torsion are given, respectively, by ∇∂∂zi∂∂zj=Γijk∂∂zk,Γijk=gkl¯∂gjl¯∂zi,Tijk=Γijk−Γjik,{\nabla }_{\tfrac{\partial }{\partial {z}_{i}}}\frac{\partial }{\partial {z}_{j}}={\Gamma }_{ij}^{k}\frac{\partial }{\partial {z}_{k}},\hspace{1em}{\Gamma }_{ij}^{k}={g}^{k\overline{l}}\frac{\partial {g}_{j\overline{l}}}{\partial {z}_{i}},\hspace{1em}{T}_{ij}^{k}={\Gamma }_{ij}^{k}-{\Gamma }_{ji}^{k},while the curvature tensor Rij¯kl¯{R}_{i\overline{j}k\overline{l}}by Rij¯kl¯=gpl¯∂Γikp∂z¯j.{R}_{i\overline{j}k\overline{l}}={g}_{p\overline{l}}\frac{\partial {\Gamma }_{ik}^{p}}{\partial {\overline{z}}_{j}}.For u∈C4(M)u\in {C}^{4}\left(M), we denote uij=∇j∇iu,uij¯=∇j¯∇iu.{u}_{ij}={\nabla }_{j}{\nabla }_{i}u,\hspace{1em}{u}_{i\overline{j}}={\nabla }_{\overline{j}}{\nabla }_{i}u.We have (see [4,11,31]) (4.1)uij¯l=ulj¯i+Tilpupj¯,uij¯k=uikj¯−glm¯Rkj¯im¯ul,uij¯k¯=uik¯j¯+Tjkl¯uil¯,uij¯k=ukij¯−glm¯Rij¯km¯ul+Tiklulj¯,\left\{\begin{array}{c}{u}_{i\overline{j}l}={u}_{l\overline{j}i}+{T}_{il}^{p}{u}_{p\overline{j}},\\ {u}_{i\overline{j}k}={u}_{ik\overline{j}}-{g}^{l\overline{m}}{R}_{k\overline{j}i\overline{m}}{u}_{l},\\ {u}_{i\overline{j}\overline{k}}={u}_{i\overline{k}\overline{j}}+\overline{{T}_{jk}^{l}}{u}_{i\overline{l}},\\ {u}_{i\overline{j}k}={u}_{ki\overline{j}}-{g}^{l\overline{m}}{R}_{i\overline{j}k\overline{m}}{u}_{l}+{T}_{ik}^{l}{u}_{l\overline{j}},\end{array}\right.and (4.2)uij¯kl¯=ukl¯ij¯+gpq¯(Rkl¯iq¯upj¯−Rij¯kq¯upl¯)+Tikpupj¯l¯+Tjlq¯uiq¯k−TikpTjlq¯upq¯.{u}_{i\overline{j}k\overline{l}}={u}_{k\overline{l}i\overline{j}}+{g}^{p\overline{q}}\left({R}_{k\overline{l}i\overline{q}}{u}_{p\overline{j}}-{R}_{i\overline{j}k\overline{q}}{u}_{p\overline{l}})+{T}_{ik}^{p}{u}_{p\overline{j}\overline{l}}+\overline{{T}_{jl}^{q}}{u}_{i\overline{q}k}-{T}_{ik}^{p}\overline{{T}_{jl}^{q}}{u}_{p\overline{q}}.Let Aij¯=gjp¯Xip¯{A}_{i\overline{j}}={g}^{j\overline{p}}{X}_{i\overline{p}}, λ(A)=(λ1,…,λn)\lambda \left(A)=\left({\lambda }_{1},\ldots ,{\lambda }_{n})and λ1≥⋯≥λn{\lambda }_{1}\hspace{0.33em}\ge \cdots \ge {\lambda }_{n}. For a fixed point x0∈M{x}_{0}\in M, choose a local coordinates such that Aij¯=Aii¯δij{A}_{i\overline{j}}={A}_{i\overline{i}}{\delta }_{ij}. Since λ1,…,λn{\lambda }_{1},\ldots ,{\lambda }_{n}need not be distinct at x0{x}_{0}, we will perturb χu{\chi }_{u}slightly such that λ1,…,λn{\lambda }_{1},\ldots ,{\lambda }_{n}become smooth functions near x0{x}_{0}. Let DDbe a diagonal matrix such that D11=0{D}^{11}=0and 0<D22<⋯<Dnn0\lt {D}^{22}\hspace{0.33em}\lt \cdots \lt {D}^{nn}are small, satisfying Dnn<2D22{D}^{nn}\lt 2{D}^{22}. Define the matrix A˜=A−D\widetilde{A}=A-D. At x0{x}_{0}, A˜\widetilde{A}has eigenvalues λ˜1=λ1,λ˜i=λi−Dii,i≥2.{\widetilde{\lambda }}_{1}={\lambda }_{1},\hspace{1em}{\widetilde{\lambda }}_{i}={\lambda }_{i}-{D}^{ii},\hspace{1em}i\ge 2.Lemma 4.1(4.3)λ˜1,ii¯≥Xii¯11¯+2Re(X11¯iT1i1¯)−C0λ1−C0.{\widetilde{\lambda }}_{1,i\overline{i}}\ge {X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\lambda }_{1}-{C}_{0}.ProofCommuting derivative of λ1˜\widetilde{{\lambda }_{1}}gives λ˜1,i=∂λ˜1∂A˜pq¯∂A˜pq¯∂zi=X11¯i−(D11)i,{\widetilde{\lambda }}_{1,i}=\frac{\partial {\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{p\overline{q}}}\frac{\partial {\widetilde{A}}_{p\overline{q}}}{\partial {z}_{i}}={X}_{1\overline{1}i}-{\left({D}^{11})}_{i},(4.4)λ˜1,ii¯=∂2λ˜1∂A˜rs¯∂A˜pq¯∂A˜pq¯∂zi∂A˜rs¯∂z¯i+∂λ˜1∂A˜pq¯∂2A˜pq¯∂z˜i∂zi=∑p≥2∣X1p¯i∣2+∣Xp1¯i∣2λ1−λ˜p−2∑p≥2Re((D1p)i¯Xp1¯i)+Re((Dp1)i¯X1p¯i)λ1−λ˜p+∑p≥2(D1p)i(Dp1)i¯+(Dp1)i(D1p)i¯λ1−λ˜p+X11¯ii¯+(D11)ii¯.\begin{array}{rcl}\phantom{\rule[-1.5em]{}{0ex}}{\widetilde{\lambda }}_{1,i\overline{i}}& =& \frac{{\partial }^{2}{\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{r\overline{s}}\partial {\widetilde{A}}_{p\overline{q}}}\frac{\partial {\widetilde{A}}_{p\overline{q}}}{\partial {z}_{i}}\frac{\partial {\widetilde{A}}_{r\overline{s}}}{\partial {\overline{z}}_{i}}+\frac{\partial {\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{p\overline{q}}}\frac{{\partial }^{2}{\widetilde{A}}_{p\overline{q}}}{\partial {\widetilde{z}}_{i}\partial {z}_{i}}\\ & =& \displaystyle \sum _{p\ge 2}\frac{| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}-2\displaystyle \sum _{p\ge 2}\frac{{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})+{\rm{Re}}({\left({D}^{p1})}_{\overline{i}}{X}_{1\overline{p}i})}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+\displaystyle \sum _{p\ge 2}\frac{{\left({D}^{1p})}_{i}{\left({D}^{p1})}_{\overline{i}}+{\left({D}^{p1})}_{i}{\left({D}^{1p})}_{\overline{i}}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+{X}_{1\overline{1}i\overline{i}}+{\left({D}^{11})}_{i\overline{i}}.\end{array}λ(A)∈Γ1\lambda \left(A)\in {\Gamma }_{1}implies that ∣λp∣<(n−1)λ1,p≥2| {\lambda }_{p}| \lt \left(n-1){\lambda }_{1},p\ge 2. If the matrix DDis sufficiently small, then ∣λ˜p∣<(n−1)λ1,p≥2| {\widetilde{\lambda }}_{p}| \lt \left(n-1){\lambda }_{1},p\ge 2, which means that 1nλ1≤1λ1−λ˜p≤1Dpp.\frac{1}{n{\lambda }_{1}}\le \frac{1}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}\le \frac{1}{{D}^{pp}}.We are trying to bound λ1{\lambda }_{1}from mentioned earlier, so we can assume λ1>1{\lambda }_{1}\gt 1. Hence, (4.5)∑p≥2(D1p)i(Dp1)i¯+(Dp1)i(D1p)i¯λ1−λ˜p+(D11)ii¯≥−C0.\sum _{p\ge 2}\frac{{\left({D}^{1p})}_{i}{\left({D}^{p1})}_{\overline{i}}+{\left({D}^{p1})}_{i}{\left({D}^{1p})}_{\overline{i}}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+{\left({D}^{11})}_{i\overline{i}}\ge -{C}_{0}.From here on, C0{C}_{0}will always denote such a constant, which depends on the given data and may vary from line to line. Using 2Re((D1p)i¯Xp1¯i)≤12∣Xp1¯i∣2+C0,2{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})\le \frac{1}{2}| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}+{C}_{0},we have (4.6)∑p≥2∣X1p¯i∣2+∣Xp1¯i∣2λ1−λ˜p−2∑p≥2Re((D1p)i¯Xp1¯i)+Re((Dp1)i¯X1p¯i)λ1−λ˜p≥12nλ1∑p≥2(∣X1p¯i∣2+∣Xp1¯i∣2)−C0.\sum _{p\ge 2}\frac{| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}-2\sum _{p\ge 2}\frac{{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})+{\rm{Re}}({\left({D}^{p1})}_{\overline{i}}{X}_{1\overline{p}i})}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}\ge \frac{1}{2n{\lambda }_{1}}\sum _{p\ge 2}(| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2})-{C}_{0}.From (4.2), we obtain (4.7)u11¯ii¯=uii¯11¯+Rii¯1p¯up1¯−R11¯ip¯upi¯+T1ipup1¯i¯+T1ip¯u1p¯i−T1ipT1iq¯upq¯.{u}_{1\overline{1}i\overline{i}}={u}_{i\overline{i}1\overline{1}}+{R}_{i\overline{i}1\overline{p}}{u}_{p\overline{1}}-{R}_{1\overline{1}i\overline{p}}{u}_{p\overline{i}}+{T}_{1i}^{p}{u}_{p\overline{1}\overline{i}}+\overline{{T}_{1i}^{p}}{u}_{1\overline{p}i}-{T}_{1i}^{p}\overline{{T}_{1i}^{q}}{u}_{p\overline{q}}.From this, we have (4.8)X11¯ii¯=Xii¯11¯+χ11¯ii¯−χii¯11¯+Rii¯1p¯up1¯−R11¯ip¯upi¯+T1ipup1¯i¯+T1ip¯u1p¯i−T1ipT1iq¯upq¯≥Xii¯11¯+λ1Rii¯11¯−λiR11¯ii¯+2Re(X1p¯iT1ip¯)−λp∣T1ip∣2−C0≥Xii¯11¯+2Re(X11¯iT1i1¯)−12nλ1∑p≥2∣X1p¯i∣2−C0λ1−C0.\begin{array}{rcl}{X}_{1\overline{1}i\overline{i}}& =& {X}_{i\overline{i}1\overline{1}}+{\chi }_{1\overline{1}i\overline{i}}-{\chi }_{i\overline{i}1\overline{1}}+{R}_{i\overline{i}1\overline{p}}{u}_{p\overline{1}}-{R}_{1\overline{1}i\overline{p}}{u}_{p\overline{i}}+{T}_{1i}^{p}{u}_{p\overline{1}\overline{i}}+\overline{{T}_{1i}^{p}}{u}_{1\overline{p}i}-{T}_{1i}^{p}\overline{{T}_{1i}^{q}}{u}_{p\overline{q}}\\ & \ge & {X}_{i\overline{i}1\overline{1}}+{\lambda }_{1}{R}_{i\overline{i}1\overline{1}}-{\lambda }_{i}{R}_{1\overline{1}i\overline{i}}+2{\rm{Re}}({X}_{1\overline{p}i}\overline{{T}_{1i}^{p}})-{\lambda }_{p}| {T}_{1i}^{p}\hspace{-0.25em}{| }^{2}-{C}_{0}\\ & \ge & {X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-\frac{1}{2n{\lambda }_{1}}\displaystyle \sum _{p\ge 2}| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}-{C}_{0}{\lambda }_{1}-{C}_{0}.\end{array}Substituting (4.5), (4.6), and (4.8) into (4.4) gives (4.3).□4.2C2{C}^{2}estimateProposition 4.2Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. Then there is an estimate as follows: supM∣∂∂¯u∣≤C(supM∣∇u∣2+1),\mathop{\sup }\limits_{M}| \partial \overline{\partial }u| \le C\left(\mathop{\sup }\limits_{M}| \nabla u\hspace{-0.25em}{| }^{2}+1\right),where C is a uniform constant.ProofWe assume that the C{\mathcal{C}}subsolution u̲=0\underline{u}=0, since otherwise we modify the background form χ\chi . We normalize uuso that supMu=0{\sup }_{M}u=0. Consider the function (4.9)W=logλ˜1+φ(∣∇u∣2)+ψ(u).W=\log {\widetilde{\lambda }}_{1}+\varphi \left(| \nabla u\hspace{-0.25em}{| }^{2})+\psi \left(u).Here, φ(t)=−12log1−t2K,0≤t≤K−1,ψ(t)=−Elog1+t2L,−L+1≤t≤0,\begin{array}{rcl}\varphi \left(t)& =& -\frac{1}{2}\log \left(1-\frac{t}{2K}\right),\hspace{1em}0\le t\le K-1,\\ \psi \left(t)& =& -E\log \left(1+\frac{t}{2L}\right),\hspace{1em}-L+1\le t\le 0,\end{array}where K=supM∣∇u∣2+1,L=supM∣u∣+1,E=2L(C1+1),K=\mathop{\sup }\limits_{M}| \nabla u\hspace{-0.25em}{| }^{2}+1,\hspace{1em}L=\mathop{\sup }\limits_{M}| u| +1,\hspace{1em}E=2L\left({C}_{1}+1),and C1{C}_{1}is to be determined later. Direct calculation gives (4.10)0<14K≤φ′≤12K,φ″=2(φ′)2,0\lt \frac{1}{4K}\le {\varphi }^{^{\prime} }\le \frac{1}{2K},\hspace{1em}{\varphi }^{^{\prime\prime} }=2{\left({\varphi }^{^{\prime} })}^{2},and (4.11)C1+1≤−ψ′≤2(C1+1),ψ″≥4ε1−ε(ψ′)2,for allε≤14E+1.{C}_{1}+1\le -{\psi }^{^{\prime} }\le 2\left({C}_{1}+1),\hspace{1em}{\psi }^{^{\prime\prime} }\ge \frac{4\varepsilon }{1-\varepsilon }{\left({\psi }^{^{\prime} })}^{2},\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \le \frac{1}{4E+1}.Since MMis compact, WWattains its maximum at some point x0∈M{x}_{0}\in M. From now on, all the calculations will be carried out at the point x0{x}_{0}and the Einstein summation convention will be used. Calculating covariant derivatives, we obtain (4.12)0=Wi=X11¯iλ1+φ′(∣∇u∣2)i+ψ′ui−(D11)iλ1,1≤i≤n,0={W}_{i}=\frac{{X}_{1\overline{1}i}}{{\lambda }_{1}}+{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}},\hspace{1.0em}1\le i\le n,(4.13)0≥Wii¯=λ˜1,ii¯λ1−λ˜1,iλ˜1,i¯λ12+ψ′uii¯+ψ″∣ui∣2+φ′(∣∇u∣2)ii¯+φ″∣(∣∇u∣2)i∣2.0\ge {W}_{i\overline{i}}=\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}-\frac{{\widetilde{\lambda }}_{1,i}{\widetilde{\lambda }}_{1,\overline{i}}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i\overline{i}}+{\varphi }^{^{\prime\prime} }| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}.Multiplying (4.13) by Fii¯{F}^{i\overline{i}}and summing it over index iiyield (4.14)0≥Fii¯λ˜1,ii¯λ1−Fii¯∣λ˜1,i∣2λ12+ψ′Fii¯uii¯+ψ″Fii¯∣ui∣2+φ′Fii¯(∣∇u∣2)ii¯+φ″Fii¯∣(∣∇u∣2)i∣2.0\ge {F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}-{F}^{i\overline{i}}\frac{| {\widetilde{\lambda }}_{1,i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime} }{F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i\overline{i}}+{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}.We will control some terms in (4.14). Covariant differentiating equation (2.4) twice in the ∂∂z1\frac{\partial }{\partial {z}^{1}}direction and the ∂∂z¯1\frac{\partial }{\partial {\overline{z}}^{1}}direction, we have (4.15)Fii¯Xii¯1+∑l=0k−2(βl)1Fl=(βk−1)1{F}^{i\overline{i}}{X}_{i\overline{i}1}+\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1}{F}_{l}={\left({\beta }_{k-1})}_{1}and (4.16)Fij¯,pq¯Xij¯1Xpq¯1¯+FiiXii¯11¯+2Re∑l=0k−2(βl)1¯Flii¯Xii¯1+∑l=0k−2(βl)11¯Fl=(βk−1)11¯.{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}+{F}^{ii}{X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}\left(\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)+\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1\overline{1}}{F}_{l}={\left({\beta }_{k-1})}_{1\overline{1}}.Direct calculation deduces that (4.17)Fii¯Xii¯=Fii¯λi=Fkii¯λi+∑l=0k−2βlFlii¯λi=βk−1−∑l=0k−2(k−l)βlFl.{F}^{i\overline{i}}{X}_{i\overline{i}}={F}^{i\overline{i}}{\lambda }_{i}={F}_{k}^{i\overline{i}}{\lambda }_{i}+\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}{F}_{l}^{i\overline{i}}{\lambda }_{i}={\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}.From Lemmas 4.1 and (4.16), we can estimate the first term in (4.14) (4.18)Fii¯λ˜1,ii¯λ1≥1λ1Fii¯Xii¯11¯+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ=−1λ1Fij¯,pq¯Xij¯1Xpq¯1¯−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1−1λ1∑l=0k−2(βl)11¯Fl+(βk−1)11¯λ1+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ.\begin{array}{rcl}{F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}& \ge & \frac{1}{{\lambda }_{1}}{F}^{i\overline{i}}{X}_{i\overline{i}1\overline{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}\\ & =& -\frac{1}{{\lambda }_{1}}{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)-\frac{1}{{\lambda }_{1}}\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1\overline{1}}{F}_{l}\\ & & +\frac{{\left({\beta }_{k-1})}_{1\overline{1}}}{{\lambda }_{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}It is shown by Krylov in [22] that the σk−1σl1k−l−1{\left(\frac{{\sigma }_{k-1}}{{\sigma }_{l}}\right)}^{\tfrac{1}{k-l-1}}is concave in Γk−1{\Gamma }_{k-1}for 0≤l≤k−20\le l\le k-2, which means that (−Fl)−1k−l−1ii¯,jj¯Xii¯1Xjj¯1¯≤0.{\left({\left(-{F}_{l})}^{-\frac{1}{k-l-1}}\right)}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}\le 0.Direct computation gives −Flii¯,jj¯Xii¯1Xjj¯1≥k−lk−l−1(−Fl)−1∣Flii¯Xii¯1∣2,-{F}_{l}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}1}\ge \frac{k-l}{k-l-1}{\left(-{F}_{l})}^{-1}| {F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\hspace{-0.25em}{| }^{2},which yields −Fii¯,jj¯Xii¯1Xjj¯1¯λ1−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1≥∑l=0k−2k−lk−l−1βlλ1(−Fl)−1Flii¯Xii¯1+k−l−1k−l(βl)1¯βlFl2+∑l=0k−2k−l−1k−l(βl)12βlλ1Fl\begin{array}{l}-\frac{{F}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}}{{\lambda }_{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)\\ \hspace{1.0em}\ge \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l}{k-l-1}\frac{{\beta }_{l}}{{\lambda }_{1}}{\left(-{F}_{l})}^{-1}{\left|,{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}+\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{\overline{1}}}{{\beta }_{l}}{F}_{l}\right|}^{2}+\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{1}^{2}}{{\beta }_{l}{\lambda }_{1}}{F}_{l}\end{array}≥∑l=0k−2k−l−1k−l(βl)12βlλ1Fl≥−C0,\begin{array}{l}\hspace{1.0em}\ge \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{1}^{2}}{{\beta }_{l}{\lambda }_{1}}{F}_{l}\\ \hspace{1.0em}\ge -{C}_{0},\end{array}where the last inequality is given by Lemma 2.2. Noting that −Fij¯,pq¯Xij¯1Xpq¯1¯≥−Fii¯,jj¯Xii¯1Xjj¯1¯−Fi1¯,1i¯∣Xi1¯1∣2,-{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}\ge -{F}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2},we have (4.19)−1λ1Fij¯,pq¯Xij¯1Xpq¯1¯−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1≥−Fi1¯,1i¯∣Xi1¯1∣2−C0.-\frac{1}{{\lambda }_{1}}{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)\ge -{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}-{C}_{0}.Substituting (4.19) into (4.18) and by Lemma 2.2, (4.20)Fii¯λ˜1,ii¯λ1≥−Fi1¯,1i¯∣Xi1¯1∣2λ1+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ−C0.{F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}\ge -\frac{{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}.Since (4.21)X11¯i=χ11¯i+u11¯i=(χ11i−χi11+Ti1pχp1¯)+Xi1¯1−Ti11λ1,{X}_{1\overline{1}i}={\chi }_{1\overline{1}i}+{u}_{1\overline{1}i}=\left({\chi }_{11i}-{\chi }_{i11}+{T}_{i1}^{p}{\chi }_{p\overline{1}})+{X}_{i\overline{1}1}-{T}_{i1}^{1}{\lambda }_{1},we have (4.22)∣X11¯i∣2≤∣Xi1¯1∣2−2λ1Re(Xi1¯1Ti11¯)+C0(λ12+∣X11¯i∣).| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}\le | {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}-2{\lambda }_{1}{\rm{Re}}\left({X}_{i\overline{1}1}\overline{{T}_{i1}^{1}})+{C}_{0}\left({\lambda }_{1}^{2}+| {X}_{1\overline{1}i}| ).From λ˜1,i=X11¯i−(D11)i,{\widetilde{\lambda }}_{1,i}={X}_{1\overline{1}i}-{\left({D}^{11})}_{i},we estimate the second term in (4.14) (4.23)−Fii¯∣λ˜1,i∣2λ12=−Fii¯∣X11¯i∣2λ12+2λ12Fii¯Re(X11¯i(D11)i¯)−Fii¯∣(D11)i∣2λi2≥−Fii¯∣X11¯i∣2λ12−C0λ12Fii¯∣X11¯i∣−C0ℱ≥−Fii¯∣Xi1¯1∣2λ12−2λ1Fii¯Re(X11¯iT1i1¯)−C0λ12Fii¯∣X11¯i∣−C0ℱ,\begin{array}{rcl}-{F}^{i\overline{i}}\frac{| {\widetilde{\lambda }}_{1,i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+\frac{2}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}{\rm{Re}}({X}_{1\overline{1}i}{\left({D}^{11})}_{\overline{i}})-\frac{{F}^{i\overline{i}}| {\left({D}^{11})}_{i}{| }^{2}}{{\lambda }_{i}^{2}}\\ & \ge & -{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{{C}_{0}}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}| {X}_{1\overline{1}i}| -{C}_{0}{\mathcal{ {\mathcal F} }}\\ & \ge & -{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-\frac{{C}_{0}}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}| {X}_{1\overline{1}i}| -{C}_{0}{\mathcal{ {\mathcal F} }},\end{array}where the last inequality is given by (4.21) and (4.22). By (4.1), we have the identities (4.24)upii¯=uii¯p−Tipiλi+Tipqχqi¯+Rii¯pq¯uq,{u}_{pi\overline{i}}={u}_{i\overline{i}p}-{T}_{ip}^{i}{\lambda }_{i}+{T}_{ip}^{q}{\chi }_{q\overline{i}}+{R}_{i\overline{i}p\overline{q}}{u}_{q},(4.25)up¯ii¯=uii¯p¯−Tipi¯λi+Tipq¯χiq¯.{u}_{\overline{p}i\overline{i}}={u}_{i\overline{i}\overline{p}}-\overline{{T}_{ip}^{i}}{\lambda }_{i}+\overline{{T}_{ip}^{q}}{\chi }_{i\overline{q}}.It follows from (4.24) and (4.25) that (4.26)Fii¯upii¯up¯=Fii¯uii¯pup¯−Fii¯Tipiλiup¯+Fii¯Tipqχqi¯up¯+Fii¯Rii¯pq¯uqup¯=−Fii¯χii¯pup¯−∑l=0k−2(βl)pFlup¯+(βk−1)pup¯−Fii¯Tipiλiup¯+Fii¯Tipqχqi¯up¯+Fii¯Rii¯pq¯uqup¯≥−C0K12ℱ+K12+K12+K12ℱ+Kℱ−C0K12Fii¯λi,\begin{array}{rcl}{F}^{i\overline{i}}{u}_{pi\overline{i}}{u}_{\overline{p}}& =& {F}^{i\overline{i}}{u}_{i\overline{i}p}{u}_{\overline{p}}-{F}^{i\overline{i}}{T}_{ip}^{i}{\lambda }_{i}{u}_{\overline{p}}+{F}^{i\overline{i}}{T}_{ip}^{q}{\chi }_{q\overline{i}}{u}_{\overline{p}}+{F}^{i\overline{i}}{R}_{i\overline{i}p\overline{q}}{u}_{q}{u}_{\overline{p}}\\ & =& -{F}^{i\overline{i}}{\chi }_{i\overline{i}p}{u}_{\overline{p}}-\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{p}{F}_{l}{u}_{\overline{p}}+{\left({\beta }_{k-1})}_{p}{u}_{\overline{p}}-{F}^{i\overline{i}}{T}_{ip}^{i}{\lambda }_{i}{u}_{\overline{p}}+{F}^{i\overline{i}}{T}_{ip}^{q}{\chi }_{q\overline{i}}{u}_{\overline{p}}+{F}^{i\overline{i}}{R}_{i\overline{i}p\overline{q}}{u}_{q}{u}_{\overline{p}}\\ & \ge & -{C}_{0}\left({K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}+{K}^{\tfrac{1}{2}}+{K}^{\tfrac{1}{2}}+{K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}+K{\mathcal{ {\mathcal F} }}\right)-{C}_{0}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}{\lambda }_{i},\end{array}where the second equality is given by (4.15) and the last inequality given by Lemma 2.2. From (4.16) and K12≤14+K{K}^{\tfrac{1}{2}}\le \frac{1}{4}+K, we obtain (4.27)−C0K12Fii¯λi=−C0K12βk−1−∑l=0k−2(k−l)βlFl≥−C0−C0K.-{C}_{0}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}{\lambda }_{i}=-{C}_{0}{K}^{\tfrac{1}{2}}\left({\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}\right)\ge -{C}_{0}-{C}_{0}K.Noting φ′≥14K\varphi ^{\prime} \ge \frac{1}{4K}, we substitute (4.27) into (4.26), (4.28)φ′Fii¯upii¯up¯≥−C0ℱ−C0Kℱ−C0−C0K.\varphi ^{\prime} {F}^{i\overline{i}}{u}_{pi\overline{i}}{u}_{\overline{p}}\ge -{C}_{0}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}.The same estimate also holds for φ′Fii¯up¯ii¯up\varphi ^{\prime} {F}^{i\overline{i}}{u}_{\overline{p}i\overline{i}}{u}_{p}. From (4.28), we can estimate the fifth term in (4.14) (4.29)φ′Fii¯(∣∇u∣2)ii¯=φ′Fii¯(upii¯up¯+up¯ii¯up)+φ′Fii¯∑p(∣upi∣2+∣up¯i∣2)≥−C0ℱ−C0Kℱ−C0−C0K+14KFii¯∑p(∣upi∣2+∣up¯i∣2).\begin{array}{rcl}\varphi ^{\prime} {F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i\overline{i}}& =& \varphi ^{\prime} {F}^{i\overline{i}}\left({u}_{pi\overline{i}}{u}_{\overline{p}}+{u}_{\overline{p}i\overline{i}}{u}_{p})+\varphi ^{\prime} {F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})\\ & \ge & -{C}_{0}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}+\frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2}).\end{array}Substituting (4.20), (4.23), and (4.29) into (4.14) (4.30)0≥−Fi1¯,1i¯∣Xi1¯1∣2λ1−Fii¯∣Xi1¯1∣2λ12−C0λ1Fii¯∣X11¯i∣λ1+14KFii¯∑p(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯+ψ″Fii¯∣ui∣2+φ″Fii¯∣(∣∇u∣2)i∣2−C0ℱ−C0−C0Kℱ−C0K.\begin{array}{rcl}0& \ge & \frac{-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}-{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}+\frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})\\ & & +{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-{C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}.\end{array}We set (4.31)δ=min11+4E,12,\delta =\min \left\{\frac{1}{1+4E},\frac{1}{2}\right\},where 11+4E=11+8L(C1+1),C1=1+1KC0θ,\frac{1}{1+4E}=\frac{1}{1+8L\left({C}_{1}+1)},\hspace{1em}{C}_{1}=\left(1+\frac{1}{K}\right)\frac{{C}_{0}}{\theta },with θ\theta in Lemma 2.6. Then we have two cases to consider.Case 1 λn<−δλ1{\lambda }_{n}\lt -\delta {\lambda }_{1}. By using the critical point condition (4.12), we obtain (4.33)−Fii¯∣X11¯i∣2λ12=−Fii¯∣φ′(∣∇u∣2)i+ψ′ui−(D11)iλ1∣2≥−2(φ′)2Fii¯∣(∣∇u∣2)i∣2−2Fii¯∣ψ′ui−(D11)iλ1∣2≥−2(φ′)2Fii¯∣(∣∇u∣2)i∣2−4∣ψ′∣2Kℱ−C0ℱ.\begin{array}{rcl}-\frac{{F}^{i\overline{i}}| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -{F}^{i\overline{i}}| {\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-2{F}^{i\overline{i}}| {\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-4| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}It follows from (4.17) that (4.34)ψ′Fii¯uii=ψ′Fii¯(Xii−χii)≥ψ′βk−1−∑l=0k−2(k−l)βlFl−C0ℱ≥ψ′(1+ℱ)C0.{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{ii}={\psi }^{^{\prime} }{F}^{i\overline{i}}\left({X}_{ii}-{\chi }_{ii})\ge {\psi }^{^{\prime} }\left({\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}-{C}_{0}{\mathcal{ {\mathcal F} }}\right)\ge {\psi }^{^{\prime} }\left(1+{\mathcal{ {\mathcal F} }}){C}_{0}.By the fact that ∣X11¯i∣λ1=−φ′(upiup¯+upup¯i)−ψ′ui+(D11)iλ1,\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}=-\varphi ^{\prime} \left({u}_{pi}{u}_{\overline{p}}+{u}_{p}{u}_{\overline{p}i})-\psi ^{\prime} {u}_{i}+\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}},we have (4.35)−C0λ1Fii¯∣X11¯i∣λ1≥−C0λ1K−12Fii¯(∣upi∣+∣up¯i∣)+C0λ1ψ′K12ℱ−C0ℱ.-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}\ge -\frac{{C}_{0}}{{\lambda }_{1}}{K}^{-\tfrac{1}{2}}{F}^{i\overline{i}}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}-{C}_{0}{\mathcal{ {\mathcal F} }}.Since −Fi1¯,1¯i=Fii¯−F11¯λ1−λiandλ1≥⋯≥λn,-{F}^{i\overline{1},\overline{1}i}=\frac{{F}^{i\overline{i}}-{F}^{1\overline{1}}}{{\lambda }_{1}-{\lambda }_{i}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\lambda }_{1}\hspace{0.33em}\ge \cdots \ge {\lambda }_{n},we have (4.36)−Fi1¯,1i¯∣Xi1¯1∣2λ1≥0.\frac{-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}\ge 0.Since φ″=2(φ′)2{\varphi }^{^{\prime\prime} }=2{\left(\varphi ^{\prime} )}^{2}and ψ″>0{\psi }^{^{\prime\prime} }\gt 0, substituting (4.32), (4.33), (4.34), and (4.35) into (4.30), we obtain (4.36)0≥14KFii¯∑p(∣upi∣2+∣up¯i∣2)−C0λ1K12Fii¯∑p(∣upi∣+∣up¯i∣)+C0λ1K12ψ′ℱ+ψ′C0(1+ℱ)−4(ψ′)2Kℱ−C0(1+ℱ)1+1K≥18KFii¯∑p(∣upi∣2+∣up¯i∣2)+C0λ1K12ψ′ℱ+ψ′C0(1+ℱ)−4(ψ′)2Kℱ−C0(1+ℱ)1+1K,\begin{array}{rcl}0& \ge & \frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})-\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\mathcal{ {\mathcal F} }}\\ & & +\psi ^{\prime} {C}_{0}\left(1+{\mathcal{ {\mathcal F} }})-4{\left(\psi ^{\prime} )}^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}\left(1+{\mathcal{ {\mathcal F} }})\left(1+\frac{1}{K}\right)\\ & \ge & \frac{1}{8K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\mathcal{ {\mathcal F} }}+\psi ^{\prime} {C}_{0}\left(1+{\mathcal{ {\mathcal F} }})-4{\left(\psi ^{\prime} )}^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}\left(1+{\mathcal{ {\mathcal F} }})\left(1+\frac{1}{K}\right),\end{array}where the last inequality is obtained by using the first term absorbing the ∣upi∣| {u}_{pi}| , ∣up¯i∣| {u}_{\overline{p}i}| terms.From Lemma 2.4, we know that ℱ{\mathcal{ {\mathcal F} }}is controlled by the uniform positive constant, which means that ℱ{\mathcal{ {\mathcal F} }}can be absorbed by C0{C}_{0}. Noticing that Fii¯∣uii¯∣2=Fii¯(λi−χii¯)2≥12Fii¯λi2−C0ℱ,andFnn¯≥ℱn≥n−k+1nk,{F}^{i\overline{i}}| {u}_{i\overline{i}}\hspace{-0.25em}{| }^{2}={F}^{i\overline{i}}{\left({\lambda }_{i}-{\chi }_{i\overline{i}})}^{2}\ge \frac{1}{2}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{\mathcal{ {\mathcal F} }},\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{F}^{n\overline{n}}\ge \frac{{\mathcal{ {\mathcal F} }}}{n}\ge \frac{n-k+1}{nk},we have 0≥116KFii¯λi2−C0K12(1+C1)−C0(1+C1)−C0(1+C1)2K−C01+1K≥(n−k+1)δ216nkKλ12−C0(1+C1)2K−C01+1K.\begin{array}{rcl}0& \ge & \frac{1}{16K}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{K}^{\tfrac{1}{2}}\left(1+{C}_{1})-{C}_{0}\left(1+{C}_{1})-{C}_{0}{\left(1+{C}_{1})}^{2}K-{C}_{0}\left(1+\frac{1}{K}\right)\\ & \ge & \frac{\left(n-k+1){\delta }^{2}}{16nkK}{\lambda }_{1}^{2}-{C}_{0}{\left(1+{C}_{1})}^{2}K-{C}_{0}\left(1+\frac{1}{K}\right).\end{array}This inequality implies λ1≤CK{\lambda }_{1}\le CK.Case 2 λn≥−δλ1{\lambda }_{n}\ge -\delta {\lambda }_{1}. Let I={i∈{1,…,n}∣Fii¯>δ−1F11¯}.I=\{i\in \left\{1,\ldots ,n\right\}| {F}^{i\overline{i}}\gt {\delta }^{-1}{F}^{1\overline{1}}\}.For those indices, which are not in II, we have (4.37)−∑i∉IFii¯∣X11¯i∣2λ12=−∑i∉IFii¯φ′(∣∇u∣2)i+ψ′ui−(D11)iλ12≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−2∑i∉IFii¯∣ψ′ui−(D11)iλ1∣2≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯−C0ℱ.\begin{array}{rcl}-\displaystyle \sum _{i\notin I}\frac{{F}^{i\overline{i}}| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{\left|,{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\right|}^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-2\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}From (4.12), (4.37), and ∣Xi1¯1∣2λ12≤∣X11¯i∣2λ12+C01+∣X11¯i∣λ1,\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}\le \frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{C}_{0}\left(1+\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}\right),we obtain (4.38)−∑i∉IFii¯∣Xi1¯1∣2λ12≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯+C0φ′∑i∉IFii¯(∣∇u∣2)i+C0ψ′∑i∉IFii¯ui−C0ℱ≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯−C0φ′K12Fii¯∑p(∣upi∣+∣up¯i∣)+C0K12ψ′δ−1F11¯−C0ℱ.\hspace{-1.25em}\begin{array}{rcl}-\displaystyle \sum _{i\notin I}\frac{{F}^{i\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}+{C}_{0}\varphi ^{\prime} \displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i}+{C}_{0}\psi ^{\prime} \displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{u}_{i}-{C}_{0}{\mathcal{ {\mathcal F} }}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}-{C}_{0}\varphi ^{\prime} {K}^{\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}Since −Fi1¯,1¯i=Fii¯−F11¯X11¯−Xii¯andλi≥λn≥−δλ1,-{F}^{i\overline{1},\overline{1}i}=\frac{{F}^{i\overline{i}}-{F}^{1\overline{1}}}{{X}_{1\overline{1}}-{X}_{i\overline{i}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\lambda }_{i}\ge {\lambda }_{n}\ge -\delta {\lambda }_{1},we have (4.39)−∑i∈IFi1¯,1¯i≥1−δ1+δ1λ1∑i∈IFii¯,-\sum _{i\in I}{F}^{i\overline{1},\overline{1}i}\ge \frac{1-\delta }{1+\delta }\frac{1}{{\lambda }_{1}}\sum _{i\in I}{F}^{i\overline{i}},which yields (4.40)−Fi1¯,1i¯∣Xi1¯1∣2λ1≥1−δ1+δ∑i∈IFii¯∣Xi1¯1∣2λ12.-\frac{{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}\ge \frac{1-\delta }{1+\delta }\sum _{i\in I}{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}.Recalling that φ″=2(φ′)2{\varphi }^{^{\prime\prime} }=2{\left(\varphi ^{\prime} )}^{2}and 0<δ≤120\lt \delta \le \frac{1}{2}, we obtain from (4.12) (4.41)∑i∈Iφ″Fii¯∣(∣∇u∣2)i∣2=2∑i∈IFii¯Xi1¯1λ1+ψ′ui−(D11)iλ1−Ti11+χ11i−χi11+Ti1pχp1¯λ12≥2∑i∈IFii¯δXi1¯1λ12−2δ1−δ(ψ′)2∣ui∣2−C0≥2δ∑i∈IFii¯Xi1¯1λ12−4δ1−δ(ψ′)2Fii¯∣ui∣2−C0ℱ.\begin{array}{rcl}\displaystyle \sum _{i\in I}{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}& =& 2\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}+\psi ^{\prime} {u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}-{T}_{i1}^{1}+\frac{{\chi }_{11i}-{\chi }_{i11}+{T}_{i1}^{p}{\chi }_{p\overline{1}}}{{\lambda }_{1}}\right|}^{2}\\ & \ge & 2\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\left(\delta \hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}\right|}^{2}-\frac{2\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}| {u}_{i}\hspace{-0.25em}{| }^{2}-{C}_{0}\right)\\ & \ge & 2\delta \displaystyle \sum _{i\in I}{F}^{i\overline{i}}\hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}\right|}^{2}-\frac{4\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}Noticing the fact that ψ″≥4ε1−ε(ψ′)2{\psi }^{^{\prime\prime} }\ge \frac{4\varepsilon }{1-\varepsilon }{\left({\psi }^{^{\prime} })}^{2}, for all ε≤14E+1=δ\varepsilon \le \frac{1}{4E+1}=\delta , we have (4.42)ψ″Fii¯∣ui∣2−4δ1−δ(ψ′)2Fii¯∣ui∣2≥0.{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}-\frac{4\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}\ge 0.Inserting (4.38), (4.40), (4.41), and (4.42) into (4.30), we deduce that (4.43)0≥14K∑pFii¯(∣upi∣2+∣up¯i∣2)+1−δ1+δ+2δ−1∑i∈IFii¯∣Xi1¯1∣2λ12+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯−C0λ1Fii¯∣X11¯i∣λ1−1+1KC0ℱ−C0−C0K−C0K−12Fii¯∑p(∣upi∣+∣up¯i∣)≥18K∑pFii¯(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯−C0λ1Fii¯∣X11¯i∣λ1−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯−1+1KC0ℱ−C0−C0K,\begin{array}{rcl}0& \ge & \frac{1}{4K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+\left(\frac{1-\delta }{1+\delta }+2\delta -1\right)\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}\\ & & -\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}\\ & & -{C}_{0}-\frac{{C}_{0}}{K}-{C}_{0}{K}^{-\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )\\ & \ge & \frac{1}{8K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}\\ & & +{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K},\end{array}where the last inequality is given by using the first term absorbing the ∣upi∣| {u}_{pi}| , ∣up¯i∣| {u}_{\overline{p}i}| terms. Combining (4.34) with (4.43), we obtain (4.44)0≥116K∑pFii¯(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯+C0λ1ψ′K12ℱ−1+1KC0ℱ−C0−C0K.\begin{array}{rcl}0& \ge & \frac{1}{16K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}\\ & & +{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}.\end{array}From Lemma 2.4, we know that ℱ{\mathcal{ {\mathcal F} }}is controlled by the uniform positive constant, which means that ℱ{\mathcal{ {\mathcal F} }}can be absorbed by C0{C}_{0}. Noticing that Fii¯∣uii¯∣2=Fii¯(λi−χii¯)2≥12Fii¯λi2−C0ℱ,{F}^{i\overline{i}}| {u}_{i\overline{i}}\hspace{-0.25em}{| }^{2}={F}^{i\overline{i}}{\left({\lambda }_{i}-{\chi }_{i\overline{i}})}^{2}\ge \frac{1}{2}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{\mathcal{ {\mathcal F} }},we obtain (4.45)0≥132K∑pFii¯λi2+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯+C0λ1ψ′K12−1+1KC0.0\ge \frac{1}{32K}\sum _{p}{F}^{i\overline{i}}{\lambda }_{i}^{2}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}-\left(1+\frac{1}{K}\right){C}_{0}.We may assume λ1≥2(C1+1)K{\lambda }_{1}\ge 2\left({C}_{1}+1)K, then C0λ1ψ′K12≥−C0K−12≥−1+1KC0.\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}\ge -{C}_{0}{K}^{-\tfrac{1}{2}}\ge -\left(1+\frac{1}{K}\right){C}_{0}.There are two cases to consider from Lemma 2.5.If (2.6) holds, then we obtain ψ′Fii¯uii¯≥(C1+1)θ(1+ℱ)≥(C1+1)θ\psi ^{\prime} {F}^{i\overline{i}}{u}_{i\overline{i}}\ge \left({C}_{1}+1)\theta \left(1+{\mathcal{ {\mathcal F} }})\ge \left({C}_{1}+1)\theta . Substituting this into (4.45) yields 0≥132KF11¯λ12+(C1+1)θ−16(C1+1)2KδF11¯−C0(C1+1)K12δF11¯−1+1KC0.0\ge \frac{1}{32K}{F}^{1\overline{1}}{\lambda }_{1}^{2}+\left({C}_{1}+1)\theta -\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }{F}^{1\overline{1}}-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta }{F}^{1\overline{1}}-\left(1+\frac{1}{K}\right){C}_{0}.Recall that C1=1+1KC0θ.{C}_{1}=\left(1+\frac{1}{K}\right)\frac{{C}_{0}}{\theta }.We then obtain 0≥132Kλ12−16(C1+1)2Kδ−C0(C1+1)K12δ,0\ge \frac{1}{32K}{\lambda }_{1}^{2}-\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta },which implies λ1≤CK{\lambda }_{1}\le CK.If (2.7) holds, then, by (4.17), 0≥132Kλ12−16(C1+1)2Kδ−C0(C1+1)K12δ−θ1+1KC0−θ(C1+1)C0.0\ge \frac{1}{32K}{\lambda }_{1}^{2}-\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta }-\theta \left(1+\frac{1}{K}\right){C}_{0}-\theta \left({C}_{1}+1){C}_{0}.This inequality again implies λ1≤CK{\lambda }_{1}\le CK.□5C1{C}^{1}estimatesIn this section, we obtain the following gradient estimate by the blowup method and the Liouville theorem as suggested by Dinew and Kolodziej [10]. The argument follows closely Proposition 5.1 in [27], so we omit the proof.Proposition 5.1Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)-form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. We normalize uusuch that supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0. Then there is an estimatesupM∣∇u∣≤C,\mathop{\sup }\limits_{M}| \nabla u| \le C,where C is a uniform constant.6Proof of main theoremFrom the standard regularity theory of uniformly elliptic partial differential equations, we can obtain the high order regularity. We refer the readers to Tosatti et al. [32]. In this section, we prove Theorem 1.2 and Corollaries 1.3 and 1.4 by the method of continuity. As explicitly shown in the proofs of the estimates up to second-order, we have to find a uniform C{\mathcal{C}}-subsolution condition for all the solution flow of the continuity method.Proof of Theorem 1.2ProofDefine ϕ̲\underline{\phi }by σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2βlσl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\hspace{2.45em}\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),where λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}. It is easy to see that (6.1)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>ϕ̲(x),\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt \underline{\phi }\left(x),where ei{e}_{i}is the iith standard basis vector in Rn{{\mathbb{R}}}^{n}. Since u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1), (6.2)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>βk−1.\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt {\beta }_{k-1}.We consider (6.3)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)ϕ̲(x)+tβk−1(x)+bt,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and bt{b}_{t}is a constant for each tt. Set T≔{t′∈[0,1]∣∃u∈C2,α(M)andbtsolving(6.3)fort∈[0,t′]}.T:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{b}_{t}\hspace{1em}{\rm{solving}}\hspace{0.33em}\left(6.3)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.As shown in [20], the continuity method works if we can guarantee (1) 0∈T0\in Tand (2) uniform C∞{C}^{\infty }estimates for all uu. When t=0,b0=0t=0,{b}_{0}=0by the uniqueness, the first requirement is naturally met. For the second requirement, we only need to show a uniform C{\mathcal{C}}-subsolution for all the solution flow. The condition (1.2) yields ϕ̲(x)≤βk−1(x).\underline{\phi }\left(x)\le {\beta }_{k-1}\left(x).At the maximum point of u−u̲u-\underline{u}, σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))≤σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2αlβl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}\le \frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\frac{{\beta }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),which means that (1−t)ϕ̲(x)+tβk−1(x)+bt≤ϕ̲(x),\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le \underline{\phi }\left(x),so bt≤0.{b}_{t}\le 0.This means that (1−t)ϕ̲(x)+tβk−1(x)+bt≤βk−1(x).\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le {\beta }_{k-1}\left(x).Then C{\mathcal{C}}-subsolution condition is uniform for all the solution flow. As a result, we have uniform C∞{C}^{\infty }estimates of uu.□Proof of Corollary 1.3ProofWe can find a smooth real function hhsatisfying that all x∈Mx\in Mh(x)≥max{ϕ̲(x),βk−1(x)}h\left(x)\ge \max \left\{\underline{\phi }\left(x),{\beta }_{k-1}\left(x)\right\}and (6.4)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>h(x),\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt h\left(x),which means that the set χ˜∈Γk−1g∣χ˜k∧ωn−k≤∑l=0k−2αl(x)χ˜l∧ωn−l+Cnk−1Cnkh(x)χ˜k−1∧ωn−k+1andχ˜−χu̲≥0\left\{\widetilde{\chi }\in {\Gamma }_{k-1}^{g}| {\widetilde{\chi }}^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\widetilde{\chi }}^{l}\wedge {\omega }^{n-l}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}h\left(x){\widetilde{\chi }}^{k-1}\wedge {\omega }^{n-k+1}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{\chi }-{\chi }_{\underline{u}}\ge 0\right\}is bounded. First, we consider (6.5)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)ϕ̲(x)+th(x)+at,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and at{a}_{t}is a constant for each tt. Set T1≔{t′∈[0,1]∣∃u∈C2,α(M)andatsolving(6.5)fort∈[0,t′]}.{T}_{1}:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{a}_{t}\hspace{0.33em}{\rm{solving}}\hspace{0.33em}\left(6.5)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.When a0=0{a}_{0}=0, 0∈T10\in {T}_{1}. At the maximum point of u−u̲u-\underline{u}, we obtain σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))≤σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2αlβl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}\le \frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\frac{{\beta }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),which means that (1−t)ϕ̲(x)+th(x)+at≤ϕ̲(x),\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t}\le \underline{\phi }\left(x),so at≤0.{a}_{t}\le 0.Obviously, (1−t)ϕ̲(x)+th(x)+at≤h(x).\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t}\le h\left(x).Second, we consider the family of equations: (6.6)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)h(x)+tβk−1(x)+bt,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)h\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and bt{b}_{t}is a constant for each tt. Set T2≔{t′∈[0,1]∣∃u∈C2,α(M)andbtsolving(6.6)fort∈[0,t′]}.{T}_{2}:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{b}_{t}\hspace{0.33em}{\rm{solving}}\hspace{0.33em}\left(6.6)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.Clearly, 0∈T20\in {T}_{2}with b0=a1{b}_{0}={a}_{1}. Integrating (6.6) on MM, we have ∫Mχk∧ωn−k=∑l=0k−2∫Mαlχul∧ωn−l+∫M(1−t)Cnk−1Cnkh+tαk−1+Cnk−1Cnkbtχuk−1∧ωn−k+1≥∑l=0k−2∫Mclχul∧ωn−l+∫Mαk−1+Cnk−1Cnkbtχuk−1∧ωn−k+1≥∑l=0k−1cl∫Mχl∧ωn−l+Cnk−1Cnkbt∫Mχk−1∧ωn−k+1≥∫Mχk∧ωn−k+Cnk−1Cnkbt∫Mχk−1∧ωn−k+1.\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}& =& \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\mathop{\displaystyle \int }\limits_{M}{\alpha }_{l}{\chi }_{u}^{l}\wedge {\omega }^{n-l}+\mathop{\displaystyle \int }\limits_{M}\left(\left(1-t)\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}h+t{\alpha }_{k-1}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\right){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\mathop{\displaystyle \int }\limits_{M}{c}_{l}{\chi }_{u}^{l}\wedge {\omega }^{n-l}+\mathop{\displaystyle \int }\limits_{M}\left({\alpha }_{k-1}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\right){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \sum }\limits_{l=0}^{k-1}{c}_{l}\mathop{\displaystyle \int }\limits_{M}{\chi }^{l}\wedge {\omega }^{n-l}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k-1}\wedge {\omega }^{n-k+1}.\end{array}The last inequality is given by the condition (1.4). Hence, bt≤0.{b}_{t}\le 0.This means that (1−t)h(x)+tβk−1(x)+bt≤h(x).\left(1-t)h\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le h\left(x).Then C{\mathcal{C}}-subsolution condition is uniform for all the solution flow. As a result, we have uniform C∞{C}^{\infty }estimates of uu.□Proof of Corollary 1.4ProofThe cone condition (1.7) is equivalent to C{\mathcal{C}}-subsolution of equation (1.6) satisfying u̲≡0\underline{u}\equiv 0. To solve equation (1.6), we consider two cases.Case 1 tan(θˆ)≥0{\rm{\tan }}\left(\hat{\theta })\ge 0.Since 2tan(θˆ)sec2(θˆ)≥02{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\ge 0, equation (1.6) is a special case of equation (1.5) and all conditions in Corollary 1.3 are satisfied. Then there exists a smooth function to solve equation (1.6) and Ωu∈Γ2g{\Omega }_{u}\in {\Gamma }_{2}^{g}. Denote the eigenvalue of gik(Ωkj¯+ukj¯){g}^{ik}\left({\Omega }_{k\bar{j}}+{u}_{k\bar{j}})as λ=(λ1,λ2,λ3)\lambda =\left({\lambda }_{1},{\lambda }_{2},{\lambda }_{3}), and λ1≥λ2≥λ3{\lambda }_{1}\ge {\lambda }_{2}\ge {\lambda }_{3}. Obviously, λ1≥λ2>0{\lambda }_{1}\ge {\lambda }_{2}\gt 0. From equation (1.6), we have λ1λ2λ3=sec2(θˆ)(λ1+λ2+λ3)+2tan(θˆ)sec2(θˆ)>0.{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}={{\rm{\sec }}}^{2}\left(\hat{\theta })\left({\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3})+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\gt 0.Hence, λ3>0{\lambda }_{3}\gt 0, which implies Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.Case 2 tan(θˆ)<0{\rm{\tan }}\left(\hat{\theta })\lt 0.In this case, θˆ∈(π2,π)\hat{\theta }\in \left(\frac{\pi }{2},\pi ). The sign of 2tan(θˆ)sec2(θˆ)2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })does not satisfy our requirement. Let Ωu=Ω˜u−sec(θˆ)ω.{\Omega }_{u}={\widetilde{\Omega }}_{u}-{\rm{\sec }}\left(\hat{\theta })\omega .Substituting Ωu{\Omega }_{u}into equation (1.6), we obtain (6.7)Ω˜u3=3sec(θˆ)Ω˜u2∧ω+2sec2(θˆ)(sec(θˆ)−tan(θˆ))ω3,{\widetilde{\Omega }}_{u}^{3}=3{\rm{\sec }}\left(\hat{\theta }){\widetilde{\Omega }}_{u}^{2}\wedge \omega +2{{\rm{\sec }}}^{2}\left(\hat{\theta })({\rm{\sec }}\left(\hat{\theta })-{\rm{\tan }}\left(\hat{\theta })){\omega }^{3},where 2sec2(θˆ)(sec(θˆ)−tan(θˆ))>02{{\rm{\sec }}}^{2}\left(\hat{\theta })({\rm{\sec }}\left(\hat{\theta })-{\rm{\tan }}\left(\hat{\theta }))\gt 0. From the cone condition (1.7), we have 3(Ω+sec(θˆ)ω)2−6sec(θˆ)(Ω+sec(θˆ)ω)∧ω>0,3{\left(\Omega +{\rm{\sec }}\left(\hat{\theta })\omega )}^{2}-6{\rm{\sec }}\left(\hat{\theta })\left(\Omega +{\rm{\sec }}\left(\hat{\theta })\omega )\wedge \omega \gt 0,which means that equation (6.7) also satisfies the cone condition. Hence, equation (6.7) satisfies all the conditions in Corollary 1.3. Then there exists a smooth function to solve equation (6.7) and Ω˜u∈Γ2g{\widetilde{\Omega }}_{u}\in {\Gamma }_{2}^{g}. Denote the eigenvalue of gik(Ωkj¯+sec(θˆ)gkj¯+ukj¯){g}^{ik}\left({\Omega }_{k\bar{j}}+{\rm{\sec }}\left(\hat{\theta }){g}_{k\bar{j}}+{u}_{k\bar{j}})as λ˜=(λ˜1,λ˜2,λ˜3)\widetilde{\lambda }=\left({\widetilde{\lambda }}_{1},{\widetilde{\lambda }}_{2},{\widetilde{\lambda }}_{3}), and λ˜1≥λ˜2≥λ˜3{\widetilde{\lambda }}_{1}\ge {\widetilde{\lambda }}_{2}\ge {\widetilde{\lambda }}_{3}. Obviously, λ˜i=λi+sec(θˆ){\widetilde{\lambda }}_{i}={\lambda }_{i}+{\rm{\sec }}\left(\hat{\theta }), for any 1≤i≤31\le i\le 3. Since λ˜∈Γ2\widetilde{\lambda }\in {\Gamma }_{2}, we obtain λ1+λ2+λ3>−3sec(θˆ).{\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3}\gt -3{\rm{\sec }}\left(\hat{\theta }).Therefore, λ1λ2λ3=sec2(θˆ)(λ1+λ2+λ3)+2tan(θˆ)sec2(θˆ)≥−3sec3(θˆ)+2tan(θˆ)sec2(θˆ)>0,{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}={{\rm{\sec }}}^{2}\left(\hat{\theta })\left({\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3})+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\ge -3{{\rm{\sec }}}^{3}\left(\hat{\theta })+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\gt 0,which implies Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Hessian equations of Krylov type on compact Hermitian manifolds

Open Mathematics , Volume 20 (1): 19 – Jan 1, 2022

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de Gruyter
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© 2022 Jundong Zhou and Yawei Chu, published by De Gruyter
ISSN
2391-5455
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2391-5455
DOI
10.1515/math-2022-0504
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1IntroductionLet (M,ω)\left(M,\omega )be a compact Kähler manifold of complex dimension nn. In 1978, Yau [1] proved the famous Calabi-Yau conjecture by solving the following complex Monge-Ampère equation on MM(ω+−1∂∂¯u)n=fωn,{\left(\omega +\sqrt{-1}\partial \overline{\partial }u)}^{n}=f{\omega }^{n},with positive function ff. There have been many generalizations of Yau’s work. One extension of Yau’s Theorem is to the case of Hermitian manifolds, which is initiated by Cherrier [2] in 1987. The Monge-Ampère equation on compact Hermitian manifolds was solved by Tosatti and Weinkove [3], building on several earlier works. See [2,4,5, 6,7] and the references therein.The complex Hessian equation can be expressed as follows: (ω+−1∂∂¯u)k∧ωn−k=fωn,2≤k≤n−1.{\left(\omega +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=f{\omega }^{n},\hspace{1.0em}2\le k\le n-1.On compact Kähler manifolds (M,ω)\left(M,\omega ), Hou [8] proved the existence of a smooth admissible solution of the complex Hessian equation by assuming the nonnegativity of the holomorphic bisectional curvature. Later, Hou et al. [9] obtained the second-order estimate without any curvature assumption. Using Hou et al.’s estimate, Dinew and Kolodziej [10] applied a blow-up argument to prove the gradient estimate and solved the complex Hessian equation on compact Kähler manifolds. The corresponding problem on Hermitian manifolds was solved by Zhang [11] and Székelyhidi [12] independently.The complex Hessian quotient equations include the complex Monge-Ampère equation and the complex Hessian equation. Let χ\chi be a real (1, 1) form, the complex Hessian quotient equations can be expressed as follows: (χ+−1∂∂¯u)k∧ωn−k=f(z)(χ+−1∂∂¯u)l∧ωn−l,1≤l<k≤n,z∈M.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=f\left(z){\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{l}\wedge {\omega }^{n-l},\hspace{1.0em}1\le l\lt k\le n,z\in M.When f(z)f\left(z)is constant, one special case is the so-called Donaldson equation [13]: (χ+−1∂∂¯u)n=c(χ+−1∂∂¯u)n−1∧ωk.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}=c{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-1}\wedge {\omega }^{k}.After some progresses made in [14,15, 16,17], Song and Weinokove [18] solved the Donaldson equation on closed Kähler manifolds via JJ-flow. Fang et al. [19] extended the Donaldson equation to (χ+−1∂∂¯u)n=ck(χ+−1∂∂¯u)n−k∧ωk{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}={c}_{k}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-k}\wedge {\omega }^{k}and solved this equation on closed Kähler manifolds by assuming a cone condition. When f(z)f\left(z)is not constant, analogous results were obtained by Sun [20,21] on compact Hermitian manifolds.In this article, we are concerned with Hessian equations of Krylov type in the form of the linear combinations of the Hessian, which can be written as follows: (1.1)(χ+−1∂∂¯u)k∧ωn−k=∑l=0k−1αl(χ+−1∂∂¯u)l∧ωn−l,2≤k≤n.{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{l}\wedge {\omega }^{n-l},\hspace{1.0em}2\le k\le n.The Dirichlet problem of (1.1) on (k−1)\left(k-1)-convex domain Ω\Omega in Rn{{\mathbb{R}}}^{n}was first studied by Krylov [22] about 20 years ago. He observed that if αl(x)≥0{\alpha }_{l}\left(x)\ge 0for 0≤l≤k−10\le l\le k-1, the natural admissible cone to make (1.1) elliptic is also the Γk{\Gamma }_{k}-cone, which is the same as the kk-Hessian equation case, where Γk={λ∈Rn∣σ1(λ)>0,…,σk(λ)>0}.{\Gamma }_{k}=\left\{\lambda \in {{\mathbb{R}}}^{n}| {\sigma }_{1}\left(\lambda )\gt 0,\ldots ,{\sigma }_{k}\left(\lambda )\gt 0\right\}.Guan and Zhang [23] solved the equation of Krylov type on the problem of prescribing convex combination of area measures. Pingali [24] proved a priori estimates to the following equation in Kähler case: (χ+−1∂∂¯u)n=∑l=0n−1Cnlαl(χ+−1∂∂¯u)n−k∧ωl,{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n}=\mathop{\sum }\limits_{l=0}^{n-1}{C}_{n}^{l}{\alpha }_{l}{\left(\chi +\sqrt{-1}\partial \overline{\partial }u)}^{n-k}\wedge {\omega }^{l},where αl≥0{\alpha }_{l}\ge 0are smooth real functions such that either αl=0{\alpha }_{l}=0or αl>0{\alpha }_{l}\gt 0, and ∑l=0n−2αl>0\mathop{\sum }\limits_{l=0}^{n-2}{\alpha }_{l}\gt 0. Recently, Phong and Tô [25] solved Hessian equations of Krylov type on compact Kähler manifolds, where αl{\alpha }_{l}are non-negative constants for 0≤l≤k−10\le l\le k-1. When αl{\alpha }_{l}are non-negative smooth functions for 0≤l≤k−10\le l\le k-1, analogous results on compact Kähler manifolds are obtained by Chen [26] and Zhou [27] independently.Naturally, we want to extend this result to Hermitian manifolds. On the other hand, Zhou [27] believed that the condition on αk−1(x)>0{\alpha }_{k-1}\left(x)\gt 0is not necessary. In fact, Guan and Zhang [23] considered equation (1.1) without the sign requirement for the coefficient function αk−1(x){\alpha }_{k-1}\left(x).In this article, we mainly concern equation (1.1) on Hermitian manifold without any sign requirement for αk−1(x){\alpha }_{k-1}\left(x). Let Γk−1g{\Gamma }_{k-1}^{g}be the set of all the real (1, 1) forms, eigenvalues of which belong to Γk−1{\Gamma }_{k-1}. To ensure the ellipticity and non degeneracy of the equation in Γk−1{\Gamma }_{k-1}, we require smooth real functions αl{\alpha }_{l}to satisfy the conditions: for 0≤l≤k−20\le l\le k-2, either αl>0{\alpha }_{l}\gt 0or αl≡0{\alpha }_{l}\equiv 0, and ∑l=0k−2αl>0{\sum }_{l=0}^{k-2}{\alpha }_{l}\gt 0. Let χu=χ+−1∂∂¯u{\chi }_{u}=\chi +\sqrt{-1}\partial \bar{\partial }uand χu̲=χ+−1∂∂¯u̲{\chi }_{\underline{u}}=\chi +\sqrt{-1}\partial \bar{\partial }\underline{u}. To state our main results, we need also the following condition of C{\mathcal{C}}-subsolution, which is similar to C{\mathcal{C}}-subsolution introduced by Székelyhidi [12].Definition 1.1A smooth real function u̲\underline{u}is a C{\mathcal{C}}-subsolution to (1.1), if χu̲∈Γk−1g,{\chi }_{\underline{u}}\in {\Gamma }_{k-1}^{g},and at each point x∈Mx\in M, the set λ(χ˜)∈Γk−1∣χ˜k∧ωn−k=∑l=0k−1αl(x)χ˜l∧ωn−landχ˜−χu̲≥0\left\{\lambda \left(\widetilde{\chi })\in {\Gamma }_{k-1}| {\widetilde{\chi }}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}\left(x){\widetilde{\chi }}^{l}\wedge {\omega }^{n-l}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{\chi }-{\chi }_{\underline{u}}\ge 0\right\}is bounded.Theorem 1.2Let (M,g)\left(M,g)be a compact Hermitian manifold, χ\chi a real (1, 1) form on MM. Suppose that u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1) and at each point x∈Mx\in M, (1.2)χu̲(x)k∧ωn−k≤∑l=0k−1αl(x)χu̲(x)l∧ωn−l.{\chi }_{\underline{u}\left(x)}^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-1}{\alpha }_{l}\left(x){\chi }_{\underline{u}\left(x)}^{l}\wedge {\omega }^{n-l}.Then there exists a smooth real function uuon MMand a unique constant bbsolving(1.3)χuk∧ωn−k=∑l=0k−2αl(x)χul∧ωn−l+(αk−1+b)χuk−1∧ωn−k+1,{\chi }_{u}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\chi }_{u}^{l}\wedge {\omega }^{n-l}+\left({\alpha }_{k-1}+b){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1},with supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0and χu∈Γk−1g{\chi }_{u}\in {\Gamma }_{k-1}^{g}.Corollary 1.3Let (M,g)\left(M,g)be a compact Kähler manifold, χ\chi a closed (1,1)\left(1,\hspace{0.33em}1)-form. Suppose that u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1) and(1.4)∫Mχk∧ωn−k≤∑l=0k−1cl∫Mχ∧ωn−l,\mathop{\int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-1}{c}_{l}\mathop{\int }\limits_{M}\chi \wedge {\omega }^{n-l},where cl=infMαl{c}_{l}={\inf }_{M}{\alpha }_{l}, 0≤l≤k−10\le l\le k-1. Then there exists a smooth real function u on M and a unique constant b solving(1.5)χuk∧ωn−k=∑l=0k−2αl(x)χul∧ωn−l+(αk−1+b)χuk−1∧ωn−k+1,{\chi }_{u}^{k}\wedge {\omega }^{n-k}=\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\chi }_{u}^{l}\wedge {\omega }^{n-l}+\left({\alpha }_{k-1}+b){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1},with supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0and χu∈Γk−1g{\chi }_{u}\in {\Gamma }_{k-1}^{g}.Lately, Pingali [28] proved an existence result of the deformed Hermitian Yang-Mills equation with phase angle θˆ∈12π,32π\hat{\theta }\in \left(\frac{1}{2}\pi ,\frac{3}{2}\pi \right)on compact Kähler threefold. Let Ω\Omega be a closed (1,1)\left(1,\hspace{0.33em}1)form, Ωu=Ω+−1∂∂¯u{\Omega }_{u}=\Omega +\sqrt{-1}\partial \bar{\partial }u. From [24], the deformed Hermitian Yang-Mills equation on compact Kähler threefold can be written as follows: (1.6)Ωu3=3sec2(θˆ)Ωu∧ω2+2tan(θˆ)sec2(θˆ)ω3.{\Omega }_{u}^{3}=3{{\rm{\sec }}}^{2}\left(\hat{\theta }){\Omega }_{u}\wedge {\omega }^{2}+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta }){\omega }^{3}.As an application of Corollary 1.3, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.Corollary 1.4Let (M,g)\left(M,g)be a compact Kähler threefold, constant phase angle θˆ∈12π,32π\hat{\theta }\in \left(\frac{1}{2}\pi ,\frac{3}{2}\pi \right), and Ω\Omega a positive definite closed (1,1)\left(1,\hspace{0.33em}1)form, satisfying the following conditions: (1.7)3Ω2−3sec2(θˆ)ω2>0,3{\Omega }^{2}-3{{\rm{\sec }}}^{2}\left(\hat{\theta }){\omega }^{2}\gt 0,(1.8)∫MΩ3=3sec2(θˆ)∫MΩ∧ω2+2tan(θˆ)sec2(θˆ)∫Mω3,\mathop{\int }\limits_{M}{\Omega }^{3}=3{{\rm{\sec }}}^{2}\left(\hat{\theta })\mathop{\int }\limits_{M}\Omega \wedge {\omega }^{2}+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\mathop{\int }\limits_{M}{\omega }^{3},(1.9)Ω+sec(θˆ)ω∈Γ2g.\Omega +{\rm{\sec }}\left(\hat{\theta })\omega \in {\Gamma }_{2}^{g}.Then there exists a smooth solution to equation (1.6) with supMu=0{\sup }_{M}u=0and Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.The rest of this article is organized as follows. In Section 2, we set up some notations and provide some preliminary results. In Section 3, we give the C0{C}^{0}estimate by the Alexandroff-Bakelman-Pucci maximum principle. In Section 4, we establish the C2{C}^{2}estimate for equation (1.1) by the method of Hou et al. [9] and the C{\mathcal{C}}-subsolution condition. In Section 5, we give the gradient estimate. In Section 6, we give the proof of Theorem 1.2, Corollaries 1.3, and 1.4 by the method of continuity. Although the method is very standard in the study of elliptic PDEs, it is not easy to carry out this method on a compact Hermitian manifold. Since the essential C{\mathcal{C}}-subsolution condition depends on α0,…,αk−1{\alpha }_{0},\ldots ,{\alpha }_{k-1}, we have to find a uniform C{\mathcal{C}}-subsolution condition for the solution flow of the continuity method.2PreliminariesIn this section, we set up the notation and establish some lemmas. Let σk(λ){\sigma }_{k}\left(\lambda )denote the kkth elementary symmetric function σk(λ)=∑1≤i1<⋯<ik≤nλi1⋯λik,forλ=(λ1,…,λn)∈Rn,1≤k≤n.{\sigma }_{k}\left(\lambda )=\sum _{1\le {i}_{1}\hspace{0.33em}\lt \cdots \lt {i}_{k}\le n}{\lambda }_{{i}_{1}}\cdots {\lambda }_{{i}_{k}},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{1.0em}\lambda =\left({\lambda }_{1},\ldots ,{\lambda }_{n})\in {{\mathbb{R}}}^{n},\hspace{1.0em}1\le k\le n.For completeness, we define σ0(λ)=1{\sigma }_{0}\left(\lambda )=1and σ−1(λ)=0{\sigma }_{-1}\left(\lambda )=0. Let σk(λ∣i){\sigma }_{k}\left(\lambda | i)denote the symmetric function with λi=0{\lambda }_{i}=0and σk(λ∣ij){\sigma }_{k}\left(\lambda | ij)the symmetric function with λi=λj=0{\lambda }_{i}={\lambda }_{j}=0. Also denote by σk(A∣i){\sigma }_{k}\left(A| i)the symmetric function with AAdeleting the iith row and iith column, and σk(A∣ij){\sigma }_{k}\left(A| ij)the symmetric function with AAdeleting the iith, jjth rows and iith, jjth columns, for all 1≤i,j≤n1\le i,j\le n. In local coordinates, Xij¯=X∂∂zi,∂∂z¯i=χij¯+uij¯,X̲ij¯=χij¯+u̲ij¯.{X}_{i\overline{j}}=X\left(\frac{\partial }{\partial {z}^{i}},\frac{\partial }{\partial {\overline{z}}^{i}}\right)={\chi }_{i\overline{j}}+{u}_{i\overline{j}},\hspace{1em}{\underline{X}}_{i\overline{j}}={\chi }_{i\overline{j}}+{\underline{u}}_{i\overline{j}}.Define λ(χu)\lambda \left({\chi }_{u})as the eigenvalue set of {Xij¯}\left\{{X}_{i\overline{j}}\right\}with respect to {gij¯}\left\{{g}_{i\overline{j}}\right\}. In local coordinates, equation (1.1) can be written in the following form: (2.1)σk(λ(χu))=∑l=0k−1βl(x)σl(λ(χu)),{\sigma }_{k}\left(\lambda \left({\chi }_{u}))=\mathop{\sum }\limits_{l=0}^{k-1}{\beta }_{l}\left(x){\sigma }_{l}\left(\lambda \left({\chi }_{u})),where σl(λ(χu))Cnl=χul∧ωn−lωn,βl(x)=CnkCnlαl(x).\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{C}_{n}^{l}}=\frac{{\chi }_{u}^{l}\wedge {\omega }^{n-l}}{{\omega }^{n}},\hspace{1em}{\beta }_{l}\left(x)=\frac{{C}_{n}^{k}}{{C}_{n}^{l}}{\alpha }_{l}\left(x).Equivalently, we can rewrite equation (2.1) as follows: (2.2)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=βk−1(x).\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}={\beta }_{k-1}\left(x).Lemma 2.1[29,30] For λ∈Γm\lambda \in {\Gamma }_{m}and m>l≥0m\gt l\ge 0, r>s≥0r\gt s\ge 0, m≥rm\ge r, l≥sl\ge s, we haveσm(λ)/Cnmσl(λ)/Cnl1m−l≤σr(λ)/Cnrσs(λ)/Cns1r−s.{\left[\frac{{\sigma }_{m}\left(\lambda )\text{/}{C}_{n}^{m}}{{\sigma }_{l}\left(\lambda )\text{/}{C}_{n}^{l}}\right]}^{\tfrac{1}{m-l}}\le {\left[\frac{{\sigma }_{r}\left(\lambda )\text{/}{C}_{n}^{r}}{{\sigma }_{s}\left(\lambda )\text{/}{C}_{n}^{s}}\right]}^{\tfrac{1}{r-s}}.The following lemma is similar to Lemma 2.3 in [27], but we need to discuss it more widely, that is, λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}instead of λ(χu)∈Γk\lambda \left({\chi }_{u})\in {\Gamma }_{k}.Lemma 2.2If u∈C2(M)u\in {C}^{2}\left(M)is a solution of (2.2), λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and βl(x)>0{\beta }_{l}\left(x)\gt 0, 0≤l≤k−20\le l\le k-2, then(2.3)σl(λ)σk−1(λ)≤C(n,k,inf0≤l≤k−2βl,sup∣βk−1∣)for0≤l≤k−2.\frac{{\sigma }_{l}\left(\lambda )}{{\sigma }_{k-1}\left(\lambda )}\le C\left(n,k,\mathop{\inf }\limits_{0\le l\le k-2}{\beta }_{l},\sup | {\beta }_{k-1}| \right)\hspace{1.0em}{for}\hspace{0.33em}0\le l\le k-2.ProofIf σkσk−1≤1\frac{{\sigma }_{k}}{{\sigma }_{k-1}}\le 1, then we obtain from equation (2.2) βl(x)σlσk−1≤σkσk−1−β(x)≤1−β(x)≤C(supM∣βk−1∣),for0≤l≤k−2.{\beta }_{l}\left(x)\frac{{\sigma }_{l}}{{\sigma }_{k-1}}\le \frac{{\sigma }_{k}}{{\sigma }_{k-1}}-\beta \left(x)\le 1-\beta \left(x)\le C\left(\mathop{\sup }\limits_{M}| {\beta }_{k-1}| \right),\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}0\le l\le k-2.If σkσk−1>1\frac{{\sigma }_{k}}{{\sigma }_{k-1}}\gt 1, i.e., σk−1σk<1\frac{{\sigma }_{k-1}}{{\sigma }_{k}}\lt 1, we see from Lemma 2.1 that σlσk−1≤(Cnk)k−1−lCnl(Cnk−1)k−lσk−1σkk−1−l≤(Cnk)k−1−lCnl(Cnk−1)k−l≤C(n,k)\frac{{\sigma }_{l}}{{\sigma }_{k-1}}\le \frac{{\left({C}_{n}^{k})}^{k-1-l}{C}_{n}^{l}}{{\left({C}_{n}^{k-1})}^{k-l}}{\left(\frac{{\sigma }_{k-1}}{{\sigma }_{k}}\right)}^{k-1-l}\le \frac{{\left({C}_{n}^{k})}^{k-1-l}{C}_{n}^{l}}{{\left({C}_{n}^{k-1})}^{k-l}}\le C\left(n,k)for 0≤l≤k−20\le l\le k-2, which completes the proof of Lemma 2.2.□For any point x0∈M{x}_{0}\in M, choose a local frame such that Xij¯=δijXii¯{X}_{i\overline{j}}={\delta }_{ij}{X}_{i\overline{i}}. For the convenience of notations, we will write equation (2.2) as follows: (2.4)F(X)=Fk(X)+∑l=0k−2βlFl(X)=βk−1(x),F\left(X)={F}_{k}\left(X)+\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}{F}_{l}\left(X)={\beta }_{k-1}\left(x),where Fk(X)=σk(λ(X))σk−1(λ(X)){F}_{k}\left(X)=\frac{{\sigma }_{k}\left(\lambda \left(X))}{{\sigma }_{k-1}\left(\lambda \left(X))}and Fl(X)=−σl(λ(X))σk−1(λ(X)){F}_{l}\left(X)=-\frac{{\sigma }_{l}\left(\lambda \left(X))}{{\sigma }_{k-1}\left(\lambda \left(X))}. Let Fij¯≔∂F∂Xij¯=∂F∂λk∂λk∂Xij¯,{F}^{i\overline{j}}:= \frac{\partial F}{\partial {X}_{i\overline{j}}}=\frac{\partial F}{\partial {\lambda }_{k}}\frac{\partial {\lambda }_{k}}{\partial {X}_{i\overline{j}}},then at x0{x}_{0}, we have Fij¯=Fii¯δij.{F}^{i\overline{j}}={F}^{i\overline{i}}{\delta }_{ij}.Let ℱ≔∑iFii¯.{\mathcal{ {\mathcal F} }}:= \sum _{i}{F}^{i\overline{i}}.Lemma 2.3[23] If λ∈Γk−1\lambda \in {\Gamma }_{k-1}and αl(x)>0{\alpha }_{l}\left(x)\gt 0, 0≤l≤k−20\le l\le k-2, then the operator F is elliptic and concave in Γk−1{\Gamma }_{k-1}.From Lemma 2.4 in [27] and Lemma 2.2, we have the following lemma.Lemma 2.4If u∈C2(M)u\in {C}^{2}\left(M)is a solution of (2.4), λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, then at x0{x}_{0}, (2.5)n−k+1k≤ℱ≤C(n,k,inf0≤l≤k−2αl,sup∣αk−1∣).\frac{n-k+1}{k}\le {\mathcal{ {\mathcal F} }}\le C\left(n,k,\mathop{\inf }\limits_{0\le l\le k-2}{\alpha }_{l},\sup | {\alpha }_{k-1}| \right).Lemma 2.5Under assumptions of Theorem 1.2, there is a constant θ>0\theta \gt 0such that(2.6)Fii¯(uii¯−u̲ii¯)≤−θ(1+ℱ),{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})\le -\theta \left(1+{\mathcal{ {\mathcal F} }}),or(2.7)F11¯≥θ.{F}^{1\overline{1}}\ge \theta .ProofWithout loss of generality, we may assume that X11¯≥⋯≥Xnn¯{X}_{1\overline{1}}\hspace{0.33em}\ge \cdots \ge {X}_{n\overline{n}}. Thus, Fnn¯≥⋯≥F11¯.{F}^{n\overline{n}}\hspace{0.33em}\ge \cdots \ge {F}^{1\overline{1}}.Since u̲\underline{u}is a C{\mathcal{C}}-subsolution, if ε>0\varepsilon \gt 0is small enough, χu̲−εω{\chi }_{\underline{u}}-\varepsilon \omega still satisfies Definition 1.1. Since MMis compact, there are uniform constants N>0N\gt 0and δ>0\delta \gt 0such that (2.8)F(X˜)>βk−1+δ,F\left(\widetilde{X})\gt {\beta }_{k-1}+\delta ,where X˜=X̲−εg+N0⋯000⋯000⋯0n×n.\widetilde{X}=\underline{X}-\varepsilon g+{\left(\begin{array}{cccc}N& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ 0& 0& \cdots & 0\end{array}\right)}_{n\times n}.Direct calculation yields (2.9)Fii¯(uii¯−u̲ii¯)=Fii¯(Xii¯−X̲ii¯)=Fii¯(Xii¯−X˜ii¯)+F11¯N−εℱ.{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})={F}^{i\overline{i}}\left({X}_{i\overline{i}}-{\underline{X}}_{i\overline{i}})={F}^{i\overline{i}}({X}_{i\overline{i}}-{\widetilde{X}}_{i\overline{i}})+{F}^{1\overline{1}}N-\varepsilon {\mathcal{ {\mathcal F} }}.Since FFis concave in Γk−1{\Gamma }_{k-1}, from (2.8), we obtain (2.10)∑i=1nFii¯(Xii¯−X˜ii¯)≤F(X)−F(X˜)≤−δ.\mathop{\sum }\limits_{i=1}^{n}{F}^{i\overline{i}}\left({X}_{i\overline{i}}-{\widetilde{X}}_{i\overline{i}})\le F\left(X)-F\left(\widetilde{X})\le -\delta .By substituting (2.10) into (2.9), we obtain Fii¯(uii¯−u̲ii¯)≤−δ+F11¯N−εℱ.{F}^{i\overline{i}}\left({u}_{i\overline{i}}-{\underline{u}}_{i\overline{i}})\le -\delta +{F}^{1\overline{1}}N-\varepsilon {\mathcal{ {\mathcal F} }}.Set θ=minδ2,ε,δ2N\theta =\min \left\{\frac{\delta }{2},\varepsilon ,\frac{\delta }{2N}\right\}. If F11¯N≤δ2{F}^{1\overline{1}}N\le \frac{\delta }{2}, we have (2.6); otherwise, (2.7) must be true.□3C0{C}^{0}estimateIn this section, we obtain the C0{C}^{0}estimate by using the Alexandroff-Bakelman-Pucci maximum principle and prove the following Proposition 3.1, which is similar to the approach of Székelyhidi [12].Proposition 3.1Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. We normalize u such that supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0. There is a constant C depending on the given data, such that(3.1)supM∣u∣<C.\mathop{\sup }\limits_{M}| u| \lt C.ProofTo simplify notation, we can assume u̲=0\underline{u}=0, otherwise we modify the background form χ\chi . Therefore, supMu=0{\sup }_{M}u=0. The following goal is to prove that L=infMuL={\inf }_{M}uhas a uniform lower bound. Notice that λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, so λ(χu)∈Γ1\lambda \left({\chi }_{u})\in {\Gamma }_{1}, that is, Δu=gip¯uip¯>−gip¯χip¯≥−C^.\Delta u={g}^{i\overline{p}}{u}_{i\overline{p}}\gt -{g}^{i\overline{p}}{\chi }_{i\overline{p}}\ge -\widehat{C}.Let G:M×M→RG:M\times M\to {\mathbb{R}}be the Green’s function of a Gauduchon metric conformal to gg. From [1], there is a uniform constant KKsuch that G(x,y)+K≥0,∀(x,y)∈M×M,and∫y∈MG(x,y)ωn(y)=0.G\left(x,y)+K\ge 0,\hspace{1em}\forall \left(x,y)\in M\times M,\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\mathop{\int }\limits_{y\in M}G\left(x,y){\omega }^{n}(y)=0.Since supMu=0{\sup }_{M}u=0, there is a point x0∈M{x}_{0}\in Msuch that u(x0)=0u\left({x}_{0})=0. Hence, u(x0)=∫Mudμ−∫y∈MG(x0,y)Δu(y)ωn(y)=∫Mudμ−∫y∈M(G(x0,y)+K)Δu(y)ωn(y)≤∫Mudμ+C^K,\begin{array}{rcl}u\left({x}_{0})& =& \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu -\mathop{\displaystyle \int }\limits_{y\in M}G\left({x}_{0},y)\Delta u(y){\omega }^{n}(y)\\ & =& \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu -\mathop{\displaystyle \int }\limits_{y\in M}\left(G\left({x}_{0},y)+K)\Delta u(y){\omega }^{n}(y)\\ & \le & \mathop{\displaystyle \int }\limits_{M}u{\rm{d}}\mu +\widehat{C}K,\end{array}which yields ∫M∣u∣dμ≤C^K.\mathop{\int }\limits_{M}| u| {\rm{d}}\mu \le \widehat{C}K.Next, we choose local coordinates at the minimum point of uuand L=infMu=u(0)L={\inf }_{M}u=u\left(0). Let B(1)={z:∣z∣<1}B\left(1)=\left\{z:| z| \lt 1\right\}and v=u+ε∣z∣2v=u+\varepsilon | z\hspace{-0.25em}{| }^{2}for a small ε>0\varepsilon \gt 0. From the Alexandroff-Bakelman-Pucci maximum principle, we obtain (3.2)c0ε2n≤∫Ωdet(D2v),{c}_{0}{\varepsilon }^{2n}\le \mathop{\int }\limits_{\Omega }\det \left({D}^{2}v),where Ω=x∈B(1):∣Dv(x)∣<ε2,v(y)≥v(x)+Dv(x)⋅(y−x),∀y∈B(1).\Omega =\left\{\begin{array}{c}x\in B\left(1):| Dv\left(x)| \lt \frac{\varepsilon }{2},\\ v(y)\ge v\left(x)+Dv\left(x)\cdot (y-x),\hspace{1em}\forall y\in B\left(1)\end{array}\right\}.Let λ˜−λ(χu̲)≥0\widetilde{\lambda }-\lambda \left({\chi }_{\underline{u}})\ge 0and σk(λ˜)σk−1(λ˜)−∑l=0k−2βlσl(λ˜)σk−1(λ˜)=βk−1(x).\frac{{\sigma }_{k}\left(\widetilde{\lambda })}{{\sigma }_{k-1}\left(\widetilde{\lambda })}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\widetilde{\lambda })}{{\sigma }_{k-1}\left(\widetilde{\lambda })}={\beta }_{k-1}\left(x).u̲\underline{u}is a C{\mathcal{C}}subsolution, which means that ∣λ˜∣| \widetilde{\lambda }| is bounded. Since MMis compact, there is uniform constant η>0\eta \gt 0such that λ(χu̲)−η1\lambda \left({\chi }_{\underline{u}})-\eta {\bf{1}}satisfies Definition 1.1. Since Ω\Omega is a contact set, we have D2v(x)≥0{D}^{2}v\left(x)\ge 0, for x∈Ωx\in \Omega , which implies uij¯(x)+εδij≥0{u}_{i\overline{j}}\left(x)+\varepsilon {\delta }_{ij}\ge 0. Choosing ε\varepsilon such that 0<ε≤η0\lt \varepsilon \le \eta , on Ω\Omega , we have λ(χu)−(λ(χu̲)−η1)≥λ(χu)−(λ(χu̲)−ε1)=λ(uij¯)+ε1≥0.\lambda \left({\chi }_{u})-(\lambda \left({\chi }_{\underline{u}})-\eta {\bf{1}})\ge \lambda \left({\chi }_{u})-(\lambda \left({\chi }_{\underline{u}})-\varepsilon {\bf{1}})=\lambda \left({u}_{i\overline{j}})+\varepsilon {\bf{1}}\ge 0.Since σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=βk−1(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}={\beta }_{k-1}\left(x),we obtain that ∣λ(χu)∣| \lambda \left({\chi }_{u})| is bounded, which yields ∣uij¯∣≤C| {u}_{i\overline{j}}| \le C. Then det(D2v(x))≤22ndet(vij¯)2≤C.\det \left({D}^{2}v\left(x))\le {2}^{2n}\det {\left({v}_{i\overline{j}})}^{2}\le C.From this and (3.2), we obtain (3.3)c0ε2n≤∫Ωdet(D2v)≤C⋅vol(Ω).{c}_{0}{\varepsilon }^{2n}\le \mathop{\int }\limits_{\Omega }{\rm{\det }}\left({D}^{2}v)\le C\cdot {\rm{vol}}\left(\Omega ).On the other hand, we have for x∈Ωx\in \Omega v(0)≥v(x)−Dv(x)⋅x>v(x)−ε2,v\left(0)\ge v\left(x)-Dv\left(x)\cdot x\gt v\left(x)-\frac{\varepsilon }{2},so ∣v(x)∣>∣L+ε2∣.| v\left(x)| \gt | L+\frac{\varepsilon }{2}| .Then, ∫M∣v(x)∣≥∫Ω∣v(x)∣≥∣L+ε2∣⋅vol(Ω).\mathop{\int }\limits_{M}| v\left(x)| \ge \mathop{\int }\limits_{\Omega }| v\left(x)| \ge | L+\frac{\varepsilon }{2}| \cdot {\rm{vol}}\left(\Omega ).Since ∫M∣v(x)∣\mathop{\int }\limits_{M}| v\left(x)| is uniformly bounded, this inequality contradicts (3.3) if LLis very large.□4C2{C}^{2}estimateIn this section, we establish the C2{C}^{2}estimate to equation (1.1). Our calculation is similar to that in [27], but on Hermitian manifolds, equation (1.1) are much more difficult to treat due to the torsion terms.4.1Notations and lemmaIn local coordinates z=(z1,…,zn)z=\left({z}_{1},\ldots ,{z}_{n}), the Chern connection ∇\nabla and torsion are given, respectively, by ∇∂∂zi∂∂zj=Γijk∂∂zk,Γijk=gkl¯∂gjl¯∂zi,Tijk=Γijk−Γjik,{\nabla }_{\tfrac{\partial }{\partial {z}_{i}}}\frac{\partial }{\partial {z}_{j}}={\Gamma }_{ij}^{k}\frac{\partial }{\partial {z}_{k}},\hspace{1em}{\Gamma }_{ij}^{k}={g}^{k\overline{l}}\frac{\partial {g}_{j\overline{l}}}{\partial {z}_{i}},\hspace{1em}{T}_{ij}^{k}={\Gamma }_{ij}^{k}-{\Gamma }_{ji}^{k},while the curvature tensor Rij¯kl¯{R}_{i\overline{j}k\overline{l}}by Rij¯kl¯=gpl¯∂Γikp∂z¯j.{R}_{i\overline{j}k\overline{l}}={g}_{p\overline{l}}\frac{\partial {\Gamma }_{ik}^{p}}{\partial {\overline{z}}_{j}}.For u∈C4(M)u\in {C}^{4}\left(M), we denote uij=∇j∇iu,uij¯=∇j¯∇iu.{u}_{ij}={\nabla }_{j}{\nabla }_{i}u,\hspace{1em}{u}_{i\overline{j}}={\nabla }_{\overline{j}}{\nabla }_{i}u.We have (see [4,11,31]) (4.1)uij¯l=ulj¯i+Tilpupj¯,uij¯k=uikj¯−glm¯Rkj¯im¯ul,uij¯k¯=uik¯j¯+Tjkl¯uil¯,uij¯k=ukij¯−glm¯Rij¯km¯ul+Tiklulj¯,\left\{\begin{array}{c}{u}_{i\overline{j}l}={u}_{l\overline{j}i}+{T}_{il}^{p}{u}_{p\overline{j}},\\ {u}_{i\overline{j}k}={u}_{ik\overline{j}}-{g}^{l\overline{m}}{R}_{k\overline{j}i\overline{m}}{u}_{l},\\ {u}_{i\overline{j}\overline{k}}={u}_{i\overline{k}\overline{j}}+\overline{{T}_{jk}^{l}}{u}_{i\overline{l}},\\ {u}_{i\overline{j}k}={u}_{ki\overline{j}}-{g}^{l\overline{m}}{R}_{i\overline{j}k\overline{m}}{u}_{l}+{T}_{ik}^{l}{u}_{l\overline{j}},\end{array}\right.and (4.2)uij¯kl¯=ukl¯ij¯+gpq¯(Rkl¯iq¯upj¯−Rij¯kq¯upl¯)+Tikpupj¯l¯+Tjlq¯uiq¯k−TikpTjlq¯upq¯.{u}_{i\overline{j}k\overline{l}}={u}_{k\overline{l}i\overline{j}}+{g}^{p\overline{q}}\left({R}_{k\overline{l}i\overline{q}}{u}_{p\overline{j}}-{R}_{i\overline{j}k\overline{q}}{u}_{p\overline{l}})+{T}_{ik}^{p}{u}_{p\overline{j}\overline{l}}+\overline{{T}_{jl}^{q}}{u}_{i\overline{q}k}-{T}_{ik}^{p}\overline{{T}_{jl}^{q}}{u}_{p\overline{q}}.Let Aij¯=gjp¯Xip¯{A}_{i\overline{j}}={g}^{j\overline{p}}{X}_{i\overline{p}}, λ(A)=(λ1,…,λn)\lambda \left(A)=\left({\lambda }_{1},\ldots ,{\lambda }_{n})and λ1≥⋯≥λn{\lambda }_{1}\hspace{0.33em}\ge \cdots \ge {\lambda }_{n}. For a fixed point x0∈M{x}_{0}\in M, choose a local coordinates such that Aij¯=Aii¯δij{A}_{i\overline{j}}={A}_{i\overline{i}}{\delta }_{ij}. Since λ1,…,λn{\lambda }_{1},\ldots ,{\lambda }_{n}need not be distinct at x0{x}_{0}, we will perturb χu{\chi }_{u}slightly such that λ1,…,λn{\lambda }_{1},\ldots ,{\lambda }_{n}become smooth functions near x0{x}_{0}. Let DDbe a diagonal matrix such that D11=0{D}^{11}=0and 0<D22<⋯<Dnn0\lt {D}^{22}\hspace{0.33em}\lt \cdots \lt {D}^{nn}are small, satisfying Dnn<2D22{D}^{nn}\lt 2{D}^{22}. Define the matrix A˜=A−D\widetilde{A}=A-D. At x0{x}_{0}, A˜\widetilde{A}has eigenvalues λ˜1=λ1,λ˜i=λi−Dii,i≥2.{\widetilde{\lambda }}_{1}={\lambda }_{1},\hspace{1em}{\widetilde{\lambda }}_{i}={\lambda }_{i}-{D}^{ii},\hspace{1em}i\ge 2.Lemma 4.1(4.3)λ˜1,ii¯≥Xii¯11¯+2Re(X11¯iT1i1¯)−C0λ1−C0.{\widetilde{\lambda }}_{1,i\overline{i}}\ge {X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\lambda }_{1}-{C}_{0}.ProofCommuting derivative of λ1˜\widetilde{{\lambda }_{1}}gives λ˜1,i=∂λ˜1∂A˜pq¯∂A˜pq¯∂zi=X11¯i−(D11)i,{\widetilde{\lambda }}_{1,i}=\frac{\partial {\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{p\overline{q}}}\frac{\partial {\widetilde{A}}_{p\overline{q}}}{\partial {z}_{i}}={X}_{1\overline{1}i}-{\left({D}^{11})}_{i},(4.4)λ˜1,ii¯=∂2λ˜1∂A˜rs¯∂A˜pq¯∂A˜pq¯∂zi∂A˜rs¯∂z¯i+∂λ˜1∂A˜pq¯∂2A˜pq¯∂z˜i∂zi=∑p≥2∣X1p¯i∣2+∣Xp1¯i∣2λ1−λ˜p−2∑p≥2Re((D1p)i¯Xp1¯i)+Re((Dp1)i¯X1p¯i)λ1−λ˜p+∑p≥2(D1p)i(Dp1)i¯+(Dp1)i(D1p)i¯λ1−λ˜p+X11¯ii¯+(D11)ii¯.\begin{array}{rcl}\phantom{\rule[-1.5em]{}{0ex}}{\widetilde{\lambda }}_{1,i\overline{i}}& =& \frac{{\partial }^{2}{\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{r\overline{s}}\partial {\widetilde{A}}_{p\overline{q}}}\frac{\partial {\widetilde{A}}_{p\overline{q}}}{\partial {z}_{i}}\frac{\partial {\widetilde{A}}_{r\overline{s}}}{\partial {\overline{z}}_{i}}+\frac{\partial {\widetilde{\lambda }}_{1}}{\partial {\widetilde{A}}_{p\overline{q}}}\frac{{\partial }^{2}{\widetilde{A}}_{p\overline{q}}}{\partial {\widetilde{z}}_{i}\partial {z}_{i}}\\ & =& \displaystyle \sum _{p\ge 2}\frac{| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}-2\displaystyle \sum _{p\ge 2}\frac{{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})+{\rm{Re}}({\left({D}^{p1})}_{\overline{i}}{X}_{1\overline{p}i})}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+\displaystyle \sum _{p\ge 2}\frac{{\left({D}^{1p})}_{i}{\left({D}^{p1})}_{\overline{i}}+{\left({D}^{p1})}_{i}{\left({D}^{1p})}_{\overline{i}}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+{X}_{1\overline{1}i\overline{i}}+{\left({D}^{11})}_{i\overline{i}}.\end{array}λ(A)∈Γ1\lambda \left(A)\in {\Gamma }_{1}implies that ∣λp∣<(n−1)λ1,p≥2| {\lambda }_{p}| \lt \left(n-1){\lambda }_{1},p\ge 2. If the matrix DDis sufficiently small, then ∣λ˜p∣<(n−1)λ1,p≥2| {\widetilde{\lambda }}_{p}| \lt \left(n-1){\lambda }_{1},p\ge 2, which means that 1nλ1≤1λ1−λ˜p≤1Dpp.\frac{1}{n{\lambda }_{1}}\le \frac{1}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}\le \frac{1}{{D}^{pp}}.We are trying to bound λ1{\lambda }_{1}from mentioned earlier, so we can assume λ1>1{\lambda }_{1}\gt 1. Hence, (4.5)∑p≥2(D1p)i(Dp1)i¯+(Dp1)i(D1p)i¯λ1−λ˜p+(D11)ii¯≥−C0.\sum _{p\ge 2}\frac{{\left({D}^{1p})}_{i}{\left({D}^{p1})}_{\overline{i}}+{\left({D}^{p1})}_{i}{\left({D}^{1p})}_{\overline{i}}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}+{\left({D}^{11})}_{i\overline{i}}\ge -{C}_{0}.From here on, C0{C}_{0}will always denote such a constant, which depends on the given data and may vary from line to line. Using 2Re((D1p)i¯Xp1¯i)≤12∣Xp1¯i∣2+C0,2{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})\le \frac{1}{2}| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}+{C}_{0},we have (4.6)∑p≥2∣X1p¯i∣2+∣Xp1¯i∣2λ1−λ˜p−2∑p≥2Re((D1p)i¯Xp1¯i)+Re((Dp1)i¯X1p¯i)λ1−λ˜p≥12nλ1∑p≥2(∣X1p¯i∣2+∣Xp1¯i∣2)−C0.\sum _{p\ge 2}\frac{| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}-2\sum _{p\ge 2}\frac{{\rm{Re}}({\left({D}^{1p})}_{\overline{i}}{X}_{p\overline{1}i})+{\rm{Re}}({\left({D}^{p1})}_{\overline{i}}{X}_{1\overline{p}i})}{{\lambda }_{1}-{\widetilde{\lambda }}_{p}}\ge \frac{1}{2n{\lambda }_{1}}\sum _{p\ge 2}(| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}+| {X}_{p\overline{1}i}\hspace{-0.25em}{| }^{2})-{C}_{0}.From (4.2), we obtain (4.7)u11¯ii¯=uii¯11¯+Rii¯1p¯up1¯−R11¯ip¯upi¯+T1ipup1¯i¯+T1ip¯u1p¯i−T1ipT1iq¯upq¯.{u}_{1\overline{1}i\overline{i}}={u}_{i\overline{i}1\overline{1}}+{R}_{i\overline{i}1\overline{p}}{u}_{p\overline{1}}-{R}_{1\overline{1}i\overline{p}}{u}_{p\overline{i}}+{T}_{1i}^{p}{u}_{p\overline{1}\overline{i}}+\overline{{T}_{1i}^{p}}{u}_{1\overline{p}i}-{T}_{1i}^{p}\overline{{T}_{1i}^{q}}{u}_{p\overline{q}}.From this, we have (4.8)X11¯ii¯=Xii¯11¯+χ11¯ii¯−χii¯11¯+Rii¯1p¯up1¯−R11¯ip¯upi¯+T1ipup1¯i¯+T1ip¯u1p¯i−T1ipT1iq¯upq¯≥Xii¯11¯+λ1Rii¯11¯−λiR11¯ii¯+2Re(X1p¯iT1ip¯)−λp∣T1ip∣2−C0≥Xii¯11¯+2Re(X11¯iT1i1¯)−12nλ1∑p≥2∣X1p¯i∣2−C0λ1−C0.\begin{array}{rcl}{X}_{1\overline{1}i\overline{i}}& =& {X}_{i\overline{i}1\overline{1}}+{\chi }_{1\overline{1}i\overline{i}}-{\chi }_{i\overline{i}1\overline{1}}+{R}_{i\overline{i}1\overline{p}}{u}_{p\overline{1}}-{R}_{1\overline{1}i\overline{p}}{u}_{p\overline{i}}+{T}_{1i}^{p}{u}_{p\overline{1}\overline{i}}+\overline{{T}_{1i}^{p}}{u}_{1\overline{p}i}-{T}_{1i}^{p}\overline{{T}_{1i}^{q}}{u}_{p\overline{q}}\\ & \ge & {X}_{i\overline{i}1\overline{1}}+{\lambda }_{1}{R}_{i\overline{i}1\overline{1}}-{\lambda }_{i}{R}_{1\overline{1}i\overline{i}}+2{\rm{Re}}({X}_{1\overline{p}i}\overline{{T}_{1i}^{p}})-{\lambda }_{p}| {T}_{1i}^{p}\hspace{-0.25em}{| }^{2}-{C}_{0}\\ & \ge & {X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-\frac{1}{2n{\lambda }_{1}}\displaystyle \sum _{p\ge 2}| {X}_{1\overline{p}i}\hspace{-0.25em}{| }^{2}-{C}_{0}{\lambda }_{1}-{C}_{0}.\end{array}Substituting (4.5), (4.6), and (4.8) into (4.4) gives (4.3).□4.2C2{C}^{2}estimateProposition 4.2Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. Then there is an estimate as follows: supM∣∂∂¯u∣≤C(supM∣∇u∣2+1),\mathop{\sup }\limits_{M}| \partial \overline{\partial }u| \le C\left(\mathop{\sup }\limits_{M}| \nabla u\hspace{-0.25em}{| }^{2}+1\right),where C is a uniform constant.ProofWe assume that the C{\mathcal{C}}subsolution u̲=0\underline{u}=0, since otherwise we modify the background form χ\chi . We normalize uuso that supMu=0{\sup }_{M}u=0. Consider the function (4.9)W=logλ˜1+φ(∣∇u∣2)+ψ(u).W=\log {\widetilde{\lambda }}_{1}+\varphi \left(| \nabla u\hspace{-0.25em}{| }^{2})+\psi \left(u).Here, φ(t)=−12log1−t2K,0≤t≤K−1,ψ(t)=−Elog1+t2L,−L+1≤t≤0,\begin{array}{rcl}\varphi \left(t)& =& -\frac{1}{2}\log \left(1-\frac{t}{2K}\right),\hspace{1em}0\le t\le K-1,\\ \psi \left(t)& =& -E\log \left(1+\frac{t}{2L}\right),\hspace{1em}-L+1\le t\le 0,\end{array}where K=supM∣∇u∣2+1,L=supM∣u∣+1,E=2L(C1+1),K=\mathop{\sup }\limits_{M}| \nabla u\hspace{-0.25em}{| }^{2}+1,\hspace{1em}L=\mathop{\sup }\limits_{M}| u| +1,\hspace{1em}E=2L\left({C}_{1}+1),and C1{C}_{1}is to be determined later. Direct calculation gives (4.10)0<14K≤φ′≤12K,φ″=2(φ′)2,0\lt \frac{1}{4K}\le {\varphi }^{^{\prime} }\le \frac{1}{2K},\hspace{1em}{\varphi }^{^{\prime\prime} }=2{\left({\varphi }^{^{\prime} })}^{2},and (4.11)C1+1≤−ψ′≤2(C1+1),ψ″≥4ε1−ε(ψ′)2,for allε≤14E+1.{C}_{1}+1\le -{\psi }^{^{\prime} }\le 2\left({C}_{1}+1),\hspace{1em}{\psi }^{^{\prime\prime} }\ge \frac{4\varepsilon }{1-\varepsilon }{\left({\psi }^{^{\prime} })}^{2},\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}\varepsilon \le \frac{1}{4E+1}.Since MMis compact, WWattains its maximum at some point x0∈M{x}_{0}\in M. From now on, all the calculations will be carried out at the point x0{x}_{0}and the Einstein summation convention will be used. Calculating covariant derivatives, we obtain (4.12)0=Wi=X11¯iλ1+φ′(∣∇u∣2)i+ψ′ui−(D11)iλ1,1≤i≤n,0={W}_{i}=\frac{{X}_{1\overline{1}i}}{{\lambda }_{1}}+{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}},\hspace{1.0em}1\le i\le n,(4.13)0≥Wii¯=λ˜1,ii¯λ1−λ˜1,iλ˜1,i¯λ12+ψ′uii¯+ψ″∣ui∣2+φ′(∣∇u∣2)ii¯+φ″∣(∣∇u∣2)i∣2.0\ge {W}_{i\overline{i}}=\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}-\frac{{\widetilde{\lambda }}_{1,i}{\widetilde{\lambda }}_{1,\overline{i}}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i\overline{i}}+{\varphi }^{^{\prime\prime} }| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}.Multiplying (4.13) by Fii¯{F}^{i\overline{i}}and summing it over index iiyield (4.14)0≥Fii¯λ˜1,ii¯λ1−Fii¯∣λ˜1,i∣2λ12+ψ′Fii¯uii¯+ψ″Fii¯∣ui∣2+φ′Fii¯(∣∇u∣2)ii¯+φ″Fii¯∣(∣∇u∣2)i∣2.0\ge {F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}-{F}^{i\overline{i}}\frac{| {\widetilde{\lambda }}_{1,i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime} }{F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i\overline{i}}+{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}.We will control some terms in (4.14). Covariant differentiating equation (2.4) twice in the ∂∂z1\frac{\partial }{\partial {z}^{1}}direction and the ∂∂z¯1\frac{\partial }{\partial {\overline{z}}^{1}}direction, we have (4.15)Fii¯Xii¯1+∑l=0k−2(βl)1Fl=(βk−1)1{F}^{i\overline{i}}{X}_{i\overline{i}1}+\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1}{F}_{l}={\left({\beta }_{k-1})}_{1}and (4.16)Fij¯,pq¯Xij¯1Xpq¯1¯+FiiXii¯11¯+2Re∑l=0k−2(βl)1¯Flii¯Xii¯1+∑l=0k−2(βl)11¯Fl=(βk−1)11¯.{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}+{F}^{ii}{X}_{i\overline{i}1\overline{1}}+2{\rm{Re}}\left(\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)+\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1\overline{1}}{F}_{l}={\left({\beta }_{k-1})}_{1\overline{1}}.Direct calculation deduces that (4.17)Fii¯Xii¯=Fii¯λi=Fkii¯λi+∑l=0k−2βlFlii¯λi=βk−1−∑l=0k−2(k−l)βlFl.{F}^{i\overline{i}}{X}_{i\overline{i}}={F}^{i\overline{i}}{\lambda }_{i}={F}_{k}^{i\overline{i}}{\lambda }_{i}+\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}{F}_{l}^{i\overline{i}}{\lambda }_{i}={\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}.From Lemmas 4.1 and (4.16), we can estimate the first term in (4.14) (4.18)Fii¯λ˜1,ii¯λ1≥1λ1Fii¯Xii¯11¯+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ=−1λ1Fij¯,pq¯Xij¯1Xpq¯1¯−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1−1λ1∑l=0k−2(βl)11¯Fl+(βk−1)11¯λ1+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ.\begin{array}{rcl}{F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}& \ge & \frac{1}{{\lambda }_{1}}{F}^{i\overline{i}}{X}_{i\overline{i}1\overline{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}\\ & =& -\frac{1}{{\lambda }_{1}}{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)-\frac{1}{{\lambda }_{1}}\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{1\overline{1}}{F}_{l}\\ & & +\frac{{\left({\beta }_{k-1})}_{1\overline{1}}}{{\lambda }_{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}It is shown by Krylov in [22] that the σk−1σl1k−l−1{\left(\frac{{\sigma }_{k-1}}{{\sigma }_{l}}\right)}^{\tfrac{1}{k-l-1}}is concave in Γk−1{\Gamma }_{k-1}for 0≤l≤k−20\le l\le k-2, which means that (−Fl)−1k−l−1ii¯,jj¯Xii¯1Xjj¯1¯≤0.{\left({\left(-{F}_{l})}^{-\frac{1}{k-l-1}}\right)}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}\le 0.Direct computation gives −Flii¯,jj¯Xii¯1Xjj¯1≥k−lk−l−1(−Fl)−1∣Flii¯Xii¯1∣2,-{F}_{l}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}1}\ge \frac{k-l}{k-l-1}{\left(-{F}_{l})}^{-1}| {F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\hspace{-0.25em}{| }^{2},which yields −Fii¯,jj¯Xii¯1Xjj¯1¯λ1−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1≥∑l=0k−2k−lk−l−1βlλ1(−Fl)−1Flii¯Xii¯1+k−l−1k−l(βl)1¯βlFl2+∑l=0k−2k−l−1k−l(βl)12βlλ1Fl\begin{array}{l}-\frac{{F}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}}{{\lambda }_{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)\\ \hspace{1.0em}\ge \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l}{k-l-1}\frac{{\beta }_{l}}{{\lambda }_{1}}{\left(-{F}_{l})}^{-1}{\left|,{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}+\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{\overline{1}}}{{\beta }_{l}}{F}_{l}\right|}^{2}+\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{1}^{2}}{{\beta }_{l}{\lambda }_{1}}{F}_{l}\end{array}≥∑l=0k−2k−l−1k−l(βl)12βlλ1Fl≥−C0,\begin{array}{l}\hspace{1.0em}\ge \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\frac{k-l-1}{k-l}\frac{{\left({\beta }_{l})}_{1}^{2}}{{\beta }_{l}{\lambda }_{1}}{F}_{l}\\ \hspace{1.0em}\ge -{C}_{0},\end{array}where the last inequality is given by Lemma 2.2. Noting that −Fij¯,pq¯Xij¯1Xpq¯1¯≥−Fii¯,jj¯Xii¯1Xjj¯1¯−Fi1¯,1i¯∣Xi1¯1∣2,-{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}\ge -{F}^{i\overline{i},j\overline{j}}{X}_{i\overline{i}1}{X}_{j\overline{j}\overline{1}}-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2},we have (4.19)−1λ1Fij¯,pq¯Xij¯1Xpq¯1¯−2λ1Re∑l=0k−2(βl)1¯Flii¯Xii¯1≥−Fi1¯,1i¯∣Xi1¯1∣2−C0.-\frac{1}{{\lambda }_{1}}{F}^{i\overline{j},p\overline{q}}{X}_{i\overline{j}1}{X}_{p\overline{q}\overline{1}}-\frac{2}{{\lambda }_{1}}{\rm{Re}}\left(\mathop{\sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{\overline{1}}{F}_{l}^{i\overline{i}}{X}_{i\overline{i}1}\right)\ge -{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}-{C}_{0}.Substituting (4.19) into (4.18) and by Lemma 2.2, (4.20)Fii¯λ˜1,ii¯λ1≥−Fi1¯,1i¯∣Xi1¯1∣2λ1+2λ1Fii¯Re(X11¯iT1i1¯)−C0ℱ−C0.{F}^{i\overline{i}}\frac{{\widetilde{\lambda }}_{1,i\overline{i}}}{{\lambda }_{1}}\ge -\frac{{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}+\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-{C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}.Since (4.21)X11¯i=χ11¯i+u11¯i=(χ11i−χi11+Ti1pχp1¯)+Xi1¯1−Ti11λ1,{X}_{1\overline{1}i}={\chi }_{1\overline{1}i}+{u}_{1\overline{1}i}=\left({\chi }_{11i}-{\chi }_{i11}+{T}_{i1}^{p}{\chi }_{p\overline{1}})+{X}_{i\overline{1}1}-{T}_{i1}^{1}{\lambda }_{1},we have (4.22)∣X11¯i∣2≤∣Xi1¯1∣2−2λ1Re(Xi1¯1Ti11¯)+C0(λ12+∣X11¯i∣).| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}\le | {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}-2{\lambda }_{1}{\rm{Re}}\left({X}_{i\overline{1}1}\overline{{T}_{i1}^{1}})+{C}_{0}\left({\lambda }_{1}^{2}+| {X}_{1\overline{1}i}| ).From λ˜1,i=X11¯i−(D11)i,{\widetilde{\lambda }}_{1,i}={X}_{1\overline{1}i}-{\left({D}^{11})}_{i},we estimate the second term in (4.14) (4.23)−Fii¯∣λ˜1,i∣2λ12=−Fii¯∣X11¯i∣2λ12+2λ12Fii¯Re(X11¯i(D11)i¯)−Fii¯∣(D11)i∣2λi2≥−Fii¯∣X11¯i∣2λ12−C0λ12Fii¯∣X11¯i∣−C0ℱ≥−Fii¯∣Xi1¯1∣2λ12−2λ1Fii¯Re(X11¯iT1i1¯)−C0λ12Fii¯∣X11¯i∣−C0ℱ,\begin{array}{rcl}-{F}^{i\overline{i}}\frac{| {\widetilde{\lambda }}_{1,i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+\frac{2}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}{\rm{Re}}({X}_{1\overline{1}i}{\left({D}^{11})}_{\overline{i}})-\frac{{F}^{i\overline{i}}| {\left({D}^{11})}_{i}{| }^{2}}{{\lambda }_{i}^{2}}\\ & \ge & -{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{{C}_{0}}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}| {X}_{1\overline{1}i}| -{C}_{0}{\mathcal{ {\mathcal F} }}\\ & \ge & -{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{2}{{\lambda }_{1}}{F}^{i\overline{i}}{\rm{Re}}\left({X}_{1\overline{1}i}\overline{{T}_{1i}^{1}})-\frac{{C}_{0}}{{\lambda }_{1}^{2}}{F}^{i\overline{i}}| {X}_{1\overline{1}i}| -{C}_{0}{\mathcal{ {\mathcal F} }},\end{array}where the last inequality is given by (4.21) and (4.22). By (4.1), we have the identities (4.24)upii¯=uii¯p−Tipiλi+Tipqχqi¯+Rii¯pq¯uq,{u}_{pi\overline{i}}={u}_{i\overline{i}p}-{T}_{ip}^{i}{\lambda }_{i}+{T}_{ip}^{q}{\chi }_{q\overline{i}}+{R}_{i\overline{i}p\overline{q}}{u}_{q},(4.25)up¯ii¯=uii¯p¯−Tipi¯λi+Tipq¯χiq¯.{u}_{\overline{p}i\overline{i}}={u}_{i\overline{i}\overline{p}}-\overline{{T}_{ip}^{i}}{\lambda }_{i}+\overline{{T}_{ip}^{q}}{\chi }_{i\overline{q}}.It follows from (4.24) and (4.25) that (4.26)Fii¯upii¯up¯=Fii¯uii¯pup¯−Fii¯Tipiλiup¯+Fii¯Tipqχqi¯up¯+Fii¯Rii¯pq¯uqup¯=−Fii¯χii¯pup¯−∑l=0k−2(βl)pFlup¯+(βk−1)pup¯−Fii¯Tipiλiup¯+Fii¯Tipqχqi¯up¯+Fii¯Rii¯pq¯uqup¯≥−C0K12ℱ+K12+K12+K12ℱ+Kℱ−C0K12Fii¯λi,\begin{array}{rcl}{F}^{i\overline{i}}{u}_{pi\overline{i}}{u}_{\overline{p}}& =& {F}^{i\overline{i}}{u}_{i\overline{i}p}{u}_{\overline{p}}-{F}^{i\overline{i}}{T}_{ip}^{i}{\lambda }_{i}{u}_{\overline{p}}+{F}^{i\overline{i}}{T}_{ip}^{q}{\chi }_{q\overline{i}}{u}_{\overline{p}}+{F}^{i\overline{i}}{R}_{i\overline{i}p\overline{q}}{u}_{q}{u}_{\overline{p}}\\ & =& -{F}^{i\overline{i}}{\chi }_{i\overline{i}p}{u}_{\overline{p}}-\mathop{\displaystyle \sum }\limits_{l=0}^{k-2}{\left({\beta }_{l})}_{p}{F}_{l}{u}_{\overline{p}}+{\left({\beta }_{k-1})}_{p}{u}_{\overline{p}}-{F}^{i\overline{i}}{T}_{ip}^{i}{\lambda }_{i}{u}_{\overline{p}}+{F}^{i\overline{i}}{T}_{ip}^{q}{\chi }_{q\overline{i}}{u}_{\overline{p}}+{F}^{i\overline{i}}{R}_{i\overline{i}p\overline{q}}{u}_{q}{u}_{\overline{p}}\\ & \ge & -{C}_{0}\left({K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}+{K}^{\tfrac{1}{2}}+{K}^{\tfrac{1}{2}}+{K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}+K{\mathcal{ {\mathcal F} }}\right)-{C}_{0}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}{\lambda }_{i},\end{array}where the second equality is given by (4.15) and the last inequality given by Lemma 2.2. From (4.16) and K12≤14+K{K}^{\tfrac{1}{2}}\le \frac{1}{4}+K, we obtain (4.27)−C0K12Fii¯λi=−C0K12βk−1−∑l=0k−2(k−l)βlFl≥−C0−C0K.-{C}_{0}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}{\lambda }_{i}=-{C}_{0}{K}^{\tfrac{1}{2}}\left({\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}\right)\ge -{C}_{0}-{C}_{0}K.Noting φ′≥14K\varphi ^{\prime} \ge \frac{1}{4K}, we substitute (4.27) into (4.26), (4.28)φ′Fii¯upii¯up¯≥−C0ℱ−C0Kℱ−C0−C0K.\varphi ^{\prime} {F}^{i\overline{i}}{u}_{pi\overline{i}}{u}_{\overline{p}}\ge -{C}_{0}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}.The same estimate also holds for φ′Fii¯up¯ii¯up\varphi ^{\prime} {F}^{i\overline{i}}{u}_{\overline{p}i\overline{i}}{u}_{p}. From (4.28), we can estimate the fifth term in (4.14) (4.29)φ′Fii¯(∣∇u∣2)ii¯=φ′Fii¯(upii¯up¯+up¯ii¯up)+φ′Fii¯∑p(∣upi∣2+∣up¯i∣2)≥−C0ℱ−C0Kℱ−C0−C0K+14KFii¯∑p(∣upi∣2+∣up¯i∣2).\begin{array}{rcl}\varphi ^{\prime} {F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i\overline{i}}& =& \varphi ^{\prime} {F}^{i\overline{i}}\left({u}_{pi\overline{i}}{u}_{\overline{p}}+{u}_{\overline{p}i\overline{i}}{u}_{p})+\varphi ^{\prime} {F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})\\ & \ge & -{C}_{0}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}+\frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2}).\end{array}Substituting (4.20), (4.23), and (4.29) into (4.14) (4.30)0≥−Fi1¯,1i¯∣Xi1¯1∣2λ1−Fii¯∣Xi1¯1∣2λ12−C0λ1Fii¯∣X11¯i∣λ1+14KFii¯∑p(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯+ψ″Fii¯∣ui∣2+φ″Fii¯∣(∣∇u∣2)i∣2−C0ℱ−C0−C0Kℱ−C0K.\begin{array}{rcl}0& \ge & \frac{-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}-{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}+\frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})\\ & & +{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}+{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}+{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-{C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}{\mathcal{ {\mathcal F} }}-\frac{{C}_{0}}{K}.\end{array}We set (4.31)δ=min11+4E,12,\delta =\min \left\{\frac{1}{1+4E},\frac{1}{2}\right\},where 11+4E=11+8L(C1+1),C1=1+1KC0θ,\frac{1}{1+4E}=\frac{1}{1+8L\left({C}_{1}+1)},\hspace{1em}{C}_{1}=\left(1+\frac{1}{K}\right)\frac{{C}_{0}}{\theta },with θ\theta in Lemma 2.6. Then we have two cases to consider.Case 1 λn<−δλ1{\lambda }_{n}\lt -\delta {\lambda }_{1}. By using the critical point condition (4.12), we obtain (4.33)−Fii¯∣X11¯i∣2λ12=−Fii¯∣φ′(∣∇u∣2)i+ψ′ui−(D11)iλ1∣2≥−2(φ′)2Fii¯∣(∣∇u∣2)i∣2−2Fii¯∣ψ′ui−(D11)iλ1∣2≥−2(φ′)2Fii¯∣(∣∇u∣2)i∣2−4∣ψ′∣2Kℱ−C0ℱ.\begin{array}{rcl}-\frac{{F}^{i\overline{i}}| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -{F}^{i\overline{i}}| {\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-2{F}^{i\overline{i}}| {\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-4| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}It follows from (4.17) that (4.34)ψ′Fii¯uii=ψ′Fii¯(Xii−χii)≥ψ′βk−1−∑l=0k−2(k−l)βlFl−C0ℱ≥ψ′(1+ℱ)C0.{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{ii}={\psi }^{^{\prime} }{F}^{i\overline{i}}\left({X}_{ii}-{\chi }_{ii})\ge {\psi }^{^{\prime} }\left({\beta }_{k-1}-\mathop{\sum }\limits_{l=0}^{k-2}\left(k-l){\beta }_{l}{F}_{l}-{C}_{0}{\mathcal{ {\mathcal F} }}\right)\ge {\psi }^{^{\prime} }\left(1+{\mathcal{ {\mathcal F} }}){C}_{0}.By the fact that ∣X11¯i∣λ1=−φ′(upiup¯+upup¯i)−ψ′ui+(D11)iλ1,\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}=-\varphi ^{\prime} \left({u}_{pi}{u}_{\overline{p}}+{u}_{p}{u}_{\overline{p}i})-\psi ^{\prime} {u}_{i}+\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}},we have (4.35)−C0λ1Fii¯∣X11¯i∣λ1≥−C0λ1K−12Fii¯(∣upi∣+∣up¯i∣)+C0λ1ψ′K12ℱ−C0ℱ.-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}\ge -\frac{{C}_{0}}{{\lambda }_{1}}{K}^{-\tfrac{1}{2}}{F}^{i\overline{i}}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}-{C}_{0}{\mathcal{ {\mathcal F} }}.Since −Fi1¯,1¯i=Fii¯−F11¯λ1−λiandλ1≥⋯≥λn,-{F}^{i\overline{1},\overline{1}i}=\frac{{F}^{i\overline{i}}-{F}^{1\overline{1}}}{{\lambda }_{1}-{\lambda }_{i}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\lambda }_{1}\hspace{0.33em}\ge \cdots \ge {\lambda }_{n},we have (4.36)−Fi1¯,1i¯∣Xi1¯1∣2λ1≥0.\frac{-{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}\ge 0.Since φ″=2(φ′)2{\varphi }^{^{\prime\prime} }=2{\left(\varphi ^{\prime} )}^{2}and ψ″>0{\psi }^{^{\prime\prime} }\gt 0, substituting (4.32), (4.33), (4.34), and (4.35) into (4.30), we obtain (4.36)0≥14KFii¯∑p(∣upi∣2+∣up¯i∣2)−C0λ1K12Fii¯∑p(∣upi∣+∣up¯i∣)+C0λ1K12ψ′ℱ+ψ′C0(1+ℱ)−4(ψ′)2Kℱ−C0(1+ℱ)1+1K≥18KFii¯∑p(∣upi∣2+∣up¯i∣2)+C0λ1K12ψ′ℱ+ψ′C0(1+ℱ)−4(ψ′)2Kℱ−C0(1+ℱ)1+1K,\begin{array}{rcl}0& \ge & \frac{1}{4K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})-\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\mathcal{ {\mathcal F} }}\\ & & +\psi ^{\prime} {C}_{0}\left(1+{\mathcal{ {\mathcal F} }})-4{\left(\psi ^{\prime} )}^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}\left(1+{\mathcal{ {\mathcal F} }})\left(1+\frac{1}{K}\right)\\ & \ge & \frac{1}{8K}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+\frac{{C}_{0}}{{\lambda }_{1}}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\mathcal{ {\mathcal F} }}+\psi ^{\prime} {C}_{0}\left(1+{\mathcal{ {\mathcal F} }})-4{\left(\psi ^{\prime} )}^{2}K{\mathcal{ {\mathcal F} }}-{C}_{0}\left(1+{\mathcal{ {\mathcal F} }})\left(1+\frac{1}{K}\right),\end{array}where the last inequality is obtained by using the first term absorbing the ∣upi∣| {u}_{pi}| , ∣up¯i∣| {u}_{\overline{p}i}| terms.From Lemma 2.4, we know that ℱ{\mathcal{ {\mathcal F} }}is controlled by the uniform positive constant, which means that ℱ{\mathcal{ {\mathcal F} }}can be absorbed by C0{C}_{0}. Noticing that Fii¯∣uii¯∣2=Fii¯(λi−χii¯)2≥12Fii¯λi2−C0ℱ,andFnn¯≥ℱn≥n−k+1nk,{F}^{i\overline{i}}| {u}_{i\overline{i}}\hspace{-0.25em}{| }^{2}={F}^{i\overline{i}}{\left({\lambda }_{i}-{\chi }_{i\overline{i}})}^{2}\ge \frac{1}{2}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{\mathcal{ {\mathcal F} }},\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{F}^{n\overline{n}}\ge \frac{{\mathcal{ {\mathcal F} }}}{n}\ge \frac{n-k+1}{nk},we have 0≥116KFii¯λi2−C0K12(1+C1)−C0(1+C1)−C0(1+C1)2K−C01+1K≥(n−k+1)δ216nkKλ12−C0(1+C1)2K−C01+1K.\begin{array}{rcl}0& \ge & \frac{1}{16K}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{K}^{\tfrac{1}{2}}\left(1+{C}_{1})-{C}_{0}\left(1+{C}_{1})-{C}_{0}{\left(1+{C}_{1})}^{2}K-{C}_{0}\left(1+\frac{1}{K}\right)\\ & \ge & \frac{\left(n-k+1){\delta }^{2}}{16nkK}{\lambda }_{1}^{2}-{C}_{0}{\left(1+{C}_{1})}^{2}K-{C}_{0}\left(1+\frac{1}{K}\right).\end{array}This inequality implies λ1≤CK{\lambda }_{1}\le CK.Case 2 λn≥−δλ1{\lambda }_{n}\ge -\delta {\lambda }_{1}. Let I={i∈{1,…,n}∣Fii¯>δ−1F11¯}.I=\{i\in \left\{1,\ldots ,n\right\}| {F}^{i\overline{i}}\gt {\delta }^{-1}{F}^{1\overline{1}}\}.For those indices, which are not in II, we have (4.37)−∑i∉IFii¯∣X11¯i∣2λ12=−∑i∉IFii¯φ′(∣∇u∣2)i+ψ′ui−(D11)iλ12≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−2∑i∉IFii¯∣ψ′ui−(D11)iλ1∣2≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯−C0ℱ.\begin{array}{rcl}-\displaystyle \sum _{i\notin I}\frac{{F}^{i\overline{i}}| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& =& -\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{\left|,{\varphi }^{^{\prime} }{\left(| \nabla u{| }^{2})}_{i}+{\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\right|}^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-2\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\psi }^{^{\prime} }{u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}\hspace{-0.25em}{| }^{2}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}From (4.12), (4.37), and ∣Xi1¯1∣2λ12≤∣X11¯i∣2λ12+C01+∣X11¯i∣λ1,\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}\le \frac{| {X}_{1\overline{1}i}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{C}_{0}\left(1+\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}\right),we obtain (4.38)−∑i∉IFii¯∣Xi1¯1∣2λ12≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯+C0φ′∑i∉IFii¯(∣∇u∣2)i+C0ψ′∑i∉IFii¯ui−C0ℱ≥−2(φ′)2∑i∉IFii¯∣(∣∇u∣2)i∣2−4Kδ∣ψ′∣2F11¯−C0φ′K12Fii¯∑p(∣upi∣+∣up¯i∣)+C0K12ψ′δ−1F11¯−C0ℱ.\hspace{-1.25em}\begin{array}{rcl}-\displaystyle \sum _{i\notin I}\frac{{F}^{i\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}& \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}+{C}_{0}\varphi ^{\prime} \displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{\left(| \nabla u{| }^{2})}_{i}+{C}_{0}\psi ^{\prime} \displaystyle \sum _{i\notin I}{F}^{i\overline{i}}{u}_{i}-{C}_{0}{\mathcal{ {\mathcal F} }}\\ & \ge & -2{\left({\varphi }^{^{\prime} })}^{2}\displaystyle \sum _{i\notin I}{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}-\frac{4K}{\delta }| {\psi }^{^{\prime} }\hspace{-0.25em}{| }^{2}{F}^{1\overline{1}}-{C}_{0}\varphi ^{\prime} {K}^{\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}Since −Fi1¯,1¯i=Fii¯−F11¯X11¯−Xii¯andλi≥λn≥−δλ1,-{F}^{i\overline{1},\overline{1}i}=\frac{{F}^{i\overline{i}}-{F}^{1\overline{1}}}{{X}_{1\overline{1}}-{X}_{i\overline{i}}}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}{\lambda }_{i}\ge {\lambda }_{n}\ge -\delta {\lambda }_{1},we have (4.39)−∑i∈IFi1¯,1¯i≥1−δ1+δ1λ1∑i∈IFii¯,-\sum _{i\in I}{F}^{i\overline{1},\overline{1}i}\ge \frac{1-\delta }{1+\delta }\frac{1}{{\lambda }_{1}}\sum _{i\in I}{F}^{i\overline{i}},which yields (4.40)−Fi1¯,1i¯∣Xi1¯1∣2λ1≥1−δ1+δ∑i∈IFii¯∣Xi1¯1∣2λ12.-\frac{{F}^{i\overline{1},1\overline{i}}| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}}\ge \frac{1-\delta }{1+\delta }\sum _{i\in I}{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}.Recalling that φ″=2(φ′)2{\varphi }^{^{\prime\prime} }=2{\left(\varphi ^{\prime} )}^{2}and 0<δ≤120\lt \delta \le \frac{1}{2}, we obtain from (4.12) (4.41)∑i∈Iφ″Fii¯∣(∣∇u∣2)i∣2=2∑i∈IFii¯Xi1¯1λ1+ψ′ui−(D11)iλ1−Ti11+χ11i−χi11+Ti1pχp1¯λ12≥2∑i∈IFii¯δXi1¯1λ12−2δ1−δ(ψ′)2∣ui∣2−C0≥2δ∑i∈IFii¯Xi1¯1λ12−4δ1−δ(ψ′)2Fii¯∣ui∣2−C0ℱ.\begin{array}{rcl}\displaystyle \sum _{i\in I}{\varphi }^{^{\prime\prime} }{F}^{i\overline{i}}| {\left(| \nabla u{| }^{2})}_{i}{| }^{2}& =& 2\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}+\psi ^{\prime} {u}_{i}-\frac{{\left({D}^{11})}_{i}}{{\lambda }_{1}}-{T}_{i1}^{1}+\frac{{\chi }_{11i}-{\chi }_{i11}+{T}_{i1}^{p}{\chi }_{p\overline{1}}}{{\lambda }_{1}}\right|}^{2}\\ & \ge & 2\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\left(\delta \hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}\right|}^{2}-\frac{2\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}| {u}_{i}\hspace{-0.25em}{| }^{2}-{C}_{0}\right)\\ & \ge & 2\delta \displaystyle \sum _{i\in I}{F}^{i\overline{i}}\hspace{-0.25em}{\left|,\frac{{X}_{i\overline{1}1}}{{\lambda }_{1}}\right|}^{2}-\frac{4\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}-{C}_{0}{\mathcal{ {\mathcal F} }}.\end{array}Noticing the fact that ψ″≥4ε1−ε(ψ′)2{\psi }^{^{\prime\prime} }\ge \frac{4\varepsilon }{1-\varepsilon }{\left({\psi }^{^{\prime} })}^{2}, for all ε≤14E+1=δ\varepsilon \le \frac{1}{4E+1}=\delta , we have (4.42)ψ″Fii¯∣ui∣2−4δ1−δ(ψ′)2Fii¯∣ui∣2≥0.{\psi }^{^{\prime\prime} }{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}-\frac{4\delta }{1-\delta }{\left({\psi }^{^{\prime} })}^{2}{F}^{i\overline{i}}| {u}_{i}\hspace{-0.25em}{| }^{2}\ge 0.Inserting (4.38), (4.40), (4.41), and (4.42) into (4.30), we deduce that (4.43)0≥14K∑pFii¯(∣upi∣2+∣up¯i∣2)+1−δ1+δ+2δ−1∑i∈IFii¯∣Xi1¯1∣2λ12+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯−C0λ1Fii¯∣X11¯i∣λ1−1+1KC0ℱ−C0−C0K−C0K−12Fii¯∑p(∣upi∣+∣up¯i∣)≥18K∑pFii¯(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯−C0λ1Fii¯∣X11¯i∣λ1−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯−1+1KC0ℱ−C0−C0K,\begin{array}{rcl}0& \ge & \frac{1}{4K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+\left(\frac{1-\delta }{1+\delta }+2\delta -1\right)\displaystyle \sum _{i\in I}{F}^{i\overline{i}}\frac{| {X}_{i\overline{1}1}\hspace{-0.25em}{| }^{2}}{{\lambda }_{1}^{2}}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}\\ & & -\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}\\ & & -{C}_{0}-\frac{{C}_{0}}{K}-{C}_{0}{K}^{-\tfrac{1}{2}}{F}^{i\overline{i}}\displaystyle \sum _{p}\left(| {u}_{pi}| +| {u}_{\overline{p}i}| )\\ & \ge & \frac{1}{8K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{{C}_{0}}{{\lambda }_{1}}{F}^{i\overline{i}}\frac{| {X}_{1\overline{1}i}| }{{\lambda }_{1}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}\\ & & +{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K},\end{array}where the last inequality is given by using the first term absorbing the ∣upi∣| {u}_{pi}| , ∣up¯i∣| {u}_{\overline{p}i}| terms. Combining (4.34) with (4.43), we obtain (4.44)0≥116K∑pFii¯(∣upi∣2+∣up¯i∣2)+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯+C0λ1ψ′K12ℱ−1+1KC0ℱ−C0−C0K.\begin{array}{rcl}0& \ge & \frac{1}{16K}\displaystyle \sum _{p}{F}^{i\overline{i}}(| {u}_{pi}\hspace{-0.25em}{| }^{2}+| {u}_{\overline{p}i}\hspace{-0.25em}{| }^{2})+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}\\ & & +{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}{\mathcal{ {\mathcal F} }}-\left(1+\frac{1}{K}\right){C}_{0}{\mathcal{ {\mathcal F} }}-{C}_{0}-\frac{{C}_{0}}{K}.\end{array}From Lemma 2.4, we know that ℱ{\mathcal{ {\mathcal F} }}is controlled by the uniform positive constant, which means that ℱ{\mathcal{ {\mathcal F} }}can be absorbed by C0{C}_{0}. Noticing that Fii¯∣uii¯∣2=Fii¯(λi−χii¯)2≥12Fii¯λi2−C0ℱ,{F}^{i\overline{i}}| {u}_{i\overline{i}}\hspace{-0.25em}{| }^{2}={F}^{i\overline{i}}{\left({\lambda }_{i}-{\chi }_{i\overline{i}})}^{2}\ge \frac{1}{2}{F}^{i\overline{i}}{\lambda }_{i}^{2}-{C}_{0}{\mathcal{ {\mathcal F} }},we obtain (4.45)0≥132K∑pFii¯λi2+ψ′Fii¯uii¯−4Kδ(ψ′)2F11¯+C0K12ψ′δ−1F11¯+C0λ1ψ′K12−1+1KC0.0\ge \frac{1}{32K}\sum _{p}{F}^{i\overline{i}}{\lambda }_{i}^{2}+{\psi }^{^{\prime} }{F}^{i\overline{i}}{u}_{i\overline{i}}-\frac{4K}{\delta }{\left(\psi ^{\prime} )}^{2}{F}^{1\overline{1}}+{C}_{0}{K}^{\tfrac{1}{2}}\psi ^{\prime} {\delta }^{-1}{F}^{1\overline{1}}+\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}-\left(1+\frac{1}{K}\right){C}_{0}.We may assume λ1≥2(C1+1)K{\lambda }_{1}\ge 2\left({C}_{1}+1)K, then C0λ1ψ′K12≥−C0K−12≥−1+1KC0.\frac{{C}_{0}}{{\lambda }_{1}}\psi ^{\prime} {K}^{\tfrac{1}{2}}\ge -{C}_{0}{K}^{-\tfrac{1}{2}}\ge -\left(1+\frac{1}{K}\right){C}_{0}.There are two cases to consider from Lemma 2.5.If (2.6) holds, then we obtain ψ′Fii¯uii¯≥(C1+1)θ(1+ℱ)≥(C1+1)θ\psi ^{\prime} {F}^{i\overline{i}}{u}_{i\overline{i}}\ge \left({C}_{1}+1)\theta \left(1+{\mathcal{ {\mathcal F} }})\ge \left({C}_{1}+1)\theta . Substituting this into (4.45) yields 0≥132KF11¯λ12+(C1+1)θ−16(C1+1)2KδF11¯−C0(C1+1)K12δF11¯−1+1KC0.0\ge \frac{1}{32K}{F}^{1\overline{1}}{\lambda }_{1}^{2}+\left({C}_{1}+1)\theta -\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }{F}^{1\overline{1}}-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta }{F}^{1\overline{1}}-\left(1+\frac{1}{K}\right){C}_{0}.Recall that C1=1+1KC0θ.{C}_{1}=\left(1+\frac{1}{K}\right)\frac{{C}_{0}}{\theta }.We then obtain 0≥132Kλ12−16(C1+1)2Kδ−C0(C1+1)K12δ,0\ge \frac{1}{32K}{\lambda }_{1}^{2}-\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta },which implies λ1≤CK{\lambda }_{1}\le CK.If (2.7) holds, then, by (4.17), 0≥132Kλ12−16(C1+1)2Kδ−C0(C1+1)K12δ−θ1+1KC0−θ(C1+1)C0.0\ge \frac{1}{32K}{\lambda }_{1}^{2}-\frac{16{\left({C}_{1}+1)}^{2}K}{\delta }-\frac{{C}_{0}\left({C}_{1}+1){K}^{\tfrac{1}{2}}}{\delta }-\theta \left(1+\frac{1}{K}\right){C}_{0}-\theta \left({C}_{1}+1){C}_{0}.This inequality again implies λ1≤CK{\lambda }_{1}\le CK.□5C1{C}^{1}estimatesIn this section, we obtain the following gradient estimate by the blowup method and the Liouville theorem as suggested by Dinew and Kolodziej [10]. The argument follows closely Proposition 5.1 in [27], so we omit the proof.Proposition 5.1Let αl(x)>0{\alpha }_{l}\left(x)\gt 0for 0≤l≤k−20\le l\le k-2and χ\chi be a smooth real (1,1)\left(1,\hspace{0.33em}1)-form on (M,g)\left(M,g). Assume that uuand u̲\underline{u}are solution and C{\mathcal{C}}-subsolution to (1.1) with λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}, λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}, respectively. We normalize uusuch that supM(u−u̲)=0{\sup }_{M}\left(u-\underline{u})=0. Then there is an estimatesupM∣∇u∣≤C,\mathop{\sup }\limits_{M}| \nabla u| \le C,where C is a uniform constant.6Proof of main theoremFrom the standard regularity theory of uniformly elliptic partial differential equations, we can obtain the high order regularity. We refer the readers to Tosatti et al. [32]. In this section, we prove Theorem 1.2 and Corollaries 1.3 and 1.4 by the method of continuity. As explicitly shown in the proofs of the estimates up to second-order, we have to find a uniform C{\mathcal{C}}-subsolution condition for all the solution flow of the continuity method.Proof of Theorem 1.2ProofDefine ϕ̲\underline{\phi }by σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2βlσl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\hspace{2.45em}\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),where λ(χu̲)∈Γk−1\lambda \left({\chi }_{\underline{u}})\in {\Gamma }_{k-1}. It is easy to see that (6.1)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>ϕ̲(x),\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt \underline{\phi }\left(x),where ei{e}_{i}is the iith standard basis vector in Rn{{\mathbb{R}}}^{n}. Since u̲\underline{u}is a C{\mathcal{C}}-subsolution of equation (1.1), (6.2)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>βk−1.\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt {\beta }_{k-1}.We consider (6.3)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)ϕ̲(x)+tβk−1(x)+bt,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and bt{b}_{t}is a constant for each tt. Set T≔{t′∈[0,1]∣∃u∈C2,α(M)andbtsolving(6.3)fort∈[0,t′]}.T:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{b}_{t}\hspace{1em}{\rm{solving}}\hspace{0.33em}\left(6.3)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.As shown in [20], the continuity method works if we can guarantee (1) 0∈T0\in Tand (2) uniform C∞{C}^{\infty }estimates for all uu. When t=0,b0=0t=0,{b}_{0}=0by the uniqueness, the first requirement is naturally met. For the second requirement, we only need to show a uniform C{\mathcal{C}}-subsolution for all the solution flow. The condition (1.2) yields ϕ̲(x)≤βk−1(x).\underline{\phi }\left(x)\le {\beta }_{k-1}\left(x).At the maximum point of u−u̲u-\underline{u}, σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))≤σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2αlβl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}\le \frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\frac{{\beta }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),which means that (1−t)ϕ̲(x)+tβk−1(x)+bt≤ϕ̲(x),\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le \underline{\phi }\left(x),so bt≤0.{b}_{t}\le 0.This means that (1−t)ϕ̲(x)+tβk−1(x)+bt≤βk−1(x).\left(1-t)\underline{\phi }\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le {\beta }_{k-1}\left(x).Then C{\mathcal{C}}-subsolution condition is uniform for all the solution flow. As a result, we have uniform C∞{C}^{\infty }estimates of uu.□Proof of Corollary 1.3ProofWe can find a smooth real function hhsatisfying that all x∈Mx\in Mh(x)≥max{ϕ̲(x),βk−1(x)}h\left(x)\ge \max \left\{\underline{\phi }\left(x),{\beta }_{k-1}\left(x)\right\}and (6.4)limt→∞σk(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)−∑l=0k−2βlσl(λ(χu̲)+tei)σk−1(λ(χu̲)+tei)>h(x),\mathop{\mathrm{lim}}\limits_{t\to \infty }\left(\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}})+t{e}_{i})}\right)\gt h\left(x),which means that the set χ˜∈Γk−1g∣χ˜k∧ωn−k≤∑l=0k−2αl(x)χ˜l∧ωn−l+Cnk−1Cnkh(x)χ˜k−1∧ωn−k+1andχ˜−χu̲≥0\left\{\widetilde{\chi }\in {\Gamma }_{k-1}^{g}| {\widetilde{\chi }}^{k}\wedge {\omega }^{n-k}\le \mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\left(x){\widetilde{\chi }}^{l}\wedge {\omega }^{n-l}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}h\left(x){\widetilde{\chi }}^{k-1}\wedge {\omega }^{n-k+1}\hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\widetilde{\chi }-{\chi }_{\underline{u}}\ge 0\right\}is bounded. First, we consider (6.5)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)ϕ̲(x)+th(x)+at,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and at{a}_{t}is a constant for each tt. Set T1≔{t′∈[0,1]∣∃u∈C2,α(M)andatsolving(6.5)fort∈[0,t′]}.{T}_{1}:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{a}_{t}\hspace{0.33em}{\rm{solving}}\hspace{0.33em}\left(6.5)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.When a0=0{a}_{0}=0, 0∈T10\in {T}_{1}. At the maximum point of u−u̲u-\underline{u}, we obtain σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))≤σk(λ(χu̲))σk−1(λ(χu̲))−∑l=0k−2αlβl(λ(χu̲))σk−1(λ(χu̲))=ϕ̲(x),\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}\le \frac{{\sigma }_{k}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\alpha }_{l}\frac{{\beta }_{l}\left(\lambda \left({\chi }_{\underline{u}}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{\underline{u}}))}=\underline{\phi }\left(x),which means that (1−t)ϕ̲(x)+th(x)+at≤ϕ̲(x),\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t}\le \underline{\phi }\left(x),so at≤0.{a}_{t}\le 0.Obviously, (1−t)ϕ̲(x)+th(x)+at≤h(x).\left(1-t)\underline{\phi }\left(x)+th\left(x)+{a}_{t}\le h\left(x).Second, we consider the family of equations: (6.6)σk(λ(χu))σk−1(λ(χu))−∑l=0k−2βlσl(λ(χu))σk−1(λ(χu))=(1−t)h(x)+tβk−1(x)+bt,\frac{{\sigma }_{k}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}-\mathop{\sum }\limits_{l=0}^{k-2}{\beta }_{l}\frac{{\sigma }_{l}\left(\lambda \left({\chi }_{u}))}{{\sigma }_{k-1}\left(\lambda \left({\chi }_{u}))}=\left(1-t)h\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t},where λ(χu)∈Γk−1\lambda \left({\chi }_{u})\in {\Gamma }_{k-1}and bt{b}_{t}is a constant for each tt. Set T2≔{t′∈[0,1]∣∃u∈C2,α(M)andbtsolving(6.6)fort∈[0,t′]}.{T}_{2}:= \left\{{t}^{^{\prime} }\in \left[0,1]| \exists u\in {C}^{2,\alpha }\left(M)\hspace{1em}{\rm{and}}\hspace{1em}{b}_{t}\hspace{0.33em}{\rm{solving}}\hspace{0.33em}\left(6.6)\hspace{0.33em}{\rm{for}}\hspace{0.33em}t\in \left[0,{t}^{^{\prime} }]\right\}.Clearly, 0∈T20\in {T}_{2}with b0=a1{b}_{0}={a}_{1}. Integrating (6.6) on MM, we have ∫Mχk∧ωn−k=∑l=0k−2∫Mαlχul∧ωn−l+∫M(1−t)Cnk−1Cnkh+tαk−1+Cnk−1Cnkbtχuk−1∧ωn−k+1≥∑l=0k−2∫Mclχul∧ωn−l+∫Mαk−1+Cnk−1Cnkbtχuk−1∧ωn−k+1≥∑l=0k−1cl∫Mχl∧ωn−l+Cnk−1Cnkbt∫Mχk−1∧ωn−k+1≥∫Mχk∧ωn−k+Cnk−1Cnkbt∫Mχk−1∧ωn−k+1.\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}& =& \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\mathop{\displaystyle \int }\limits_{M}{\alpha }_{l}{\chi }_{u}^{l}\wedge {\omega }^{n-l}+\mathop{\displaystyle \int }\limits_{M}\left(\left(1-t)\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}h+t{\alpha }_{k-1}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\right){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \sum }\limits_{l=0}^{k-2}\mathop{\displaystyle \int }\limits_{M}{c}_{l}{\chi }_{u}^{l}\wedge {\omega }^{n-l}+\mathop{\displaystyle \int }\limits_{M}\left({\alpha }_{k-1}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\right){\chi }_{u}^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \sum }\limits_{l=0}^{k-1}{c}_{l}\mathop{\displaystyle \int }\limits_{M}{\chi }^{l}\wedge {\omega }^{n-l}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k-1}\wedge {\omega }^{n-k+1}\\ & \ge & \mathop{\displaystyle \int }\limits_{M}{\chi }^{k}\wedge {\omega }^{n-k}+\frac{{C}_{n}^{k-1}}{{C}_{n}^{k}}{b}_{t}\mathop{\displaystyle \int }\limits_{M}{\chi }^{k-1}\wedge {\omega }^{n-k+1}.\end{array}The last inequality is given by the condition (1.4). Hence, bt≤0.{b}_{t}\le 0.This means that (1−t)h(x)+tβk−1(x)+bt≤h(x).\left(1-t)h\left(x)+t{\beta }_{k-1}\left(x)+{b}_{t}\le h\left(x).Then C{\mathcal{C}}-subsolution condition is uniform for all the solution flow. As a result, we have uniform C∞{C}^{\infty }estimates of uu.□Proof of Corollary 1.4ProofThe cone condition (1.7) is equivalent to C{\mathcal{C}}-subsolution of equation (1.6) satisfying u̲≡0\underline{u}\equiv 0. To solve equation (1.6), we consider two cases.Case 1 tan(θˆ)≥0{\rm{\tan }}\left(\hat{\theta })\ge 0.Since 2tan(θˆ)sec2(θˆ)≥02{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\ge 0, equation (1.6) is a special case of equation (1.5) and all conditions in Corollary 1.3 are satisfied. Then there exists a smooth function to solve equation (1.6) and Ωu∈Γ2g{\Omega }_{u}\in {\Gamma }_{2}^{g}. Denote the eigenvalue of gik(Ωkj¯+ukj¯){g}^{ik}\left({\Omega }_{k\bar{j}}+{u}_{k\bar{j}})as λ=(λ1,λ2,λ3)\lambda =\left({\lambda }_{1},{\lambda }_{2},{\lambda }_{3}), and λ1≥λ2≥λ3{\lambda }_{1}\ge {\lambda }_{2}\ge {\lambda }_{3}. Obviously, λ1≥λ2>0{\lambda }_{1}\ge {\lambda }_{2}\gt 0. From equation (1.6), we have λ1λ2λ3=sec2(θˆ)(λ1+λ2+λ3)+2tan(θˆ)sec2(θˆ)>0.{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}={{\rm{\sec }}}^{2}\left(\hat{\theta })\left({\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3})+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\gt 0.Hence, λ3>0{\lambda }_{3}\gt 0, which implies Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.Case 2 tan(θˆ)<0{\rm{\tan }}\left(\hat{\theta })\lt 0.In this case, θˆ∈(π2,π)\hat{\theta }\in \left(\frac{\pi }{2},\pi ). The sign of 2tan(θˆ)sec2(θˆ)2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })does not satisfy our requirement. Let Ωu=Ω˜u−sec(θˆ)ω.{\Omega }_{u}={\widetilde{\Omega }}_{u}-{\rm{\sec }}\left(\hat{\theta })\omega .Substituting Ωu{\Omega }_{u}into equation (1.6), we obtain (6.7)Ω˜u3=3sec(θˆ)Ω˜u2∧ω+2sec2(θˆ)(sec(θˆ)−tan(θˆ))ω3,{\widetilde{\Omega }}_{u}^{3}=3{\rm{\sec }}\left(\hat{\theta }){\widetilde{\Omega }}_{u}^{2}\wedge \omega +2{{\rm{\sec }}}^{2}\left(\hat{\theta })({\rm{\sec }}\left(\hat{\theta })-{\rm{\tan }}\left(\hat{\theta })){\omega }^{3},where 2sec2(θˆ)(sec(θˆ)−tan(θˆ))>02{{\rm{\sec }}}^{2}\left(\hat{\theta })({\rm{\sec }}\left(\hat{\theta })-{\rm{\tan }}\left(\hat{\theta }))\gt 0. From the cone condition (1.7), we have 3(Ω+sec(θˆ)ω)2−6sec(θˆ)(Ω+sec(θˆ)ω)∧ω>0,3{\left(\Omega +{\rm{\sec }}\left(\hat{\theta })\omega )}^{2}-6{\rm{\sec }}\left(\hat{\theta })\left(\Omega +{\rm{\sec }}\left(\hat{\theta })\omega )\wedge \omega \gt 0,which means that equation (6.7) also satisfies the cone condition. Hence, equation (6.7) satisfies all the conditions in Corollary 1.3. Then there exists a smooth function to solve equation (6.7) and Ω˜u∈Γ2g{\widetilde{\Omega }}_{u}\in {\Gamma }_{2}^{g}. Denote the eigenvalue of gik(Ωkj¯+sec(θˆ)gkj¯+ukj¯){g}^{ik}\left({\Omega }_{k\bar{j}}+{\rm{\sec }}\left(\hat{\theta }){g}_{k\bar{j}}+{u}_{k\bar{j}})as λ˜=(λ˜1,λ˜2,λ˜3)\widetilde{\lambda }=\left({\widetilde{\lambda }}_{1},{\widetilde{\lambda }}_{2},{\widetilde{\lambda }}_{3}), and λ˜1≥λ˜2≥λ˜3{\widetilde{\lambda }}_{1}\ge {\widetilde{\lambda }}_{2}\ge {\widetilde{\lambda }}_{3}. Obviously, λ˜i=λi+sec(θˆ){\widetilde{\lambda }}_{i}={\lambda }_{i}+{\rm{\sec }}\left(\hat{\theta }), for any 1≤i≤31\le i\le 3. Since λ˜∈Γ2\widetilde{\lambda }\in {\Gamma }_{2}, we obtain λ1+λ2+λ3>−3sec(θˆ).{\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3}\gt -3{\rm{\sec }}\left(\hat{\theta }).Therefore, λ1λ2λ3=sec2(θˆ)(λ1+λ2+λ3)+2tan(θˆ)sec2(θˆ)≥−3sec3(θˆ)+2tan(θˆ)sec2(θˆ)>0,{\lambda }_{1}{\lambda }_{2}{\lambda }_{3}={{\rm{\sec }}}^{2}\left(\hat{\theta })\left({\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3})+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\ge -3{{\rm{\sec }}}^{3}\left(\hat{\theta })+2{\rm{\tan }}\left(\hat{\theta }){{\rm{\sec }}}^{2}\left(\hat{\theta })\gt 0,which implies Ωu∈Γ3g{\Omega }_{u}\in {\Gamma }_{3}^{g}.□

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: Hessian equations; Hermitian manifolds; subsolution condition; 35J60; 35B45; 53C55

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