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Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian... 1Introduction and main resultsIn this paper, we consider the following nonlinear Schrödinger equations involving the fractional p-Laplacian (1.1)(−Δ)psu(x)+λV(x)u(x)p−1=u(x)q−1,x∈RN,u(x)≥0,u(x)∈Ws,p(RN),\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\left(-\Delta )}_{p}^{s}u\left(x)+\lambda V\left(x)u{\left(x)}^{p-1}=u{\left(x)}^{q-1},\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\ge 0,\hspace{1em}\hspace{1em}u\left(x)\in {W}^{s,p}\left({{\mathbb{R}}}^{N}),\end{array}\right.where λ>0\lambda \gt 0is a parameter, 1<p<q<NpN−sp1\lt p\lt q\lt \frac{Np}{N-sp}, N≥2N\ge 2, and V(x)V\left(x)is a real continuous function on RN{{\mathbb{R}}}^{N}.We are interested in the existence of ground state solutions for λ\lambda big enough, and their asymptotical behavior as λ→∞.\lambda \to \infty .As far as we know, these kinds of problems were first put forward in [1] by Bartsch and Wang, where they studied the Schrödinger equations. Under suitable conditions imposed on the potential, the loss of compactness caused by the whole space RN{{\mathbb{R}}}^{N}can be recovered when parameter λ\lambda is big enough. Then, many authors began studying the problems with potential well. A lot of results have been obtained.Bartsch and Parnet [2] also considered the nonlinear Schrödinger equation: −Δu+(a0(x)+λa(x))u=f(x,u),x∈RN,u(x)→0,∣x∣→∞,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\hspace{-0.25em}\Delta u+\left({a}_{0}\left(x)+\lambda a\left(x))u=f\left(x,u),& \hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,& \hspace{0.95em}| x| \to \infty ,\end{array}\right.where a0(x)+λa(x){a}_{0}\left(x)+\lambda a\left(x)is indefinite. By using a local linking theorem and the critical groups theory, they obtained the existence of solutions and their asymptotical behavior as λ→∞\lambda \to \infty .Xu and Chen [3] studied the following Kirchhoff problem: −a+b∫RN∣∇u∣2dxΔu+λV(x)u=f(x,u),x∈RN,u(x)∈H1(RN),\left\{\begin{array}{l}-\hspace{-0.25em}\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+\lambda V\left(x)u=f\left(x,u),\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right.where f(x,u)f\left(x,u)can be sublinear or superlinear. By using the genus theory, they obtained infinitely many negative solutions.Aleves et al. [4] dealt with the following Choquard equation: −Δu+(λa(x)+1)u=1∣x∣μ∗∣u∣p∣u∣p−2u,x∈RN,u(x)∈H1(RN),\left\{\begin{array}{l}-\hspace{-0.25em}\Delta u+\left(\lambda a\left(x)+1)u=\left(\frac{1}{| x\hspace{-0.25em}{| }^{\mu }}\ast | u\hspace{-0.25em}{| }^{p}\right)| u\hspace{-0.25em}{| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right.where μ∈(0,3)\mu \in \left(0,3), p∈(2,6−μ)p\in \left(2,6-\mu ), and the potential well Ω=⋃j=1kΩj\Omega ={\bigcup }_{j=1}^{k}{\Omega }_{j}. They proved the existence of a solution, which is nonzero on any subset Ωj{\Omega }_{j}. Furthermore, its asymptotical behavior was investigated.Zhao et al. [5] studied the Schrödinger-Poisson system allowing the potential V(x)V\left(x)changes sign −Δu+λV(x)u+K(x)ϕu=∣u∣p−2uinR3,−Δϕ=K(x)u2inR3,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\hspace{-0.25em}\Delta u+\lambda V\left(x)u+K\left(x)\phi u=| u\hspace{-0.25em}{| }^{p-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.where p∈(3,6)p\in \left(3,6)and V∈C(R3,R)V\in C\left({{\mathbb{R}}}^{3},{\mathbb{R}})are bounded from below. Using the variational method, they obtained the existence and asymptotic behavior of nontrivial solutions.For the critical problems, Clapp and Ding [6] have studied the nonlinear Schrödinger equation: −Δu+λV(x)u=μu+u2∗−1,x∈RN\hspace{-18.75em}-\Delta u+\lambda V\left(x)u=\mu u+{u}^{{2}^{\ast }-1},\hspace{1.0em}x\in {{\mathbb{R}}}^{N}for N≥4,λ,μ>0N\ge 4,\lambda ,\mu \gt 0. By using variational methods, the authors established existence and multiplicity of positive solutions, which localize near the potential well for λ\lambda large and μ\mu small.Later, the corresponding results obtained in [6] were generalized to the fractional Schrödinger equations by Niu and Tang [7], where they have studied (−Δ)su+(λV(x)−μ)u=∣u∣2s∗−2u,x∈RN,u≥0,u∈Hs(RN).\hspace{0.45em}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\left(-\Delta )}^{s}u+\left(\lambda V\left(x)-\mu )u=| u\hspace{-0.25em}{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\ge 0,\hspace{1em}u\in {H}^{s}\left({{\mathbb{R}}}^{N}).\end{array}\right.Under the linear perturbation, [6] and [7] obtained the existence of solutions and their asymptotic behavior. For the nonlinear perturbation, Alves and Barros [8] considered −Δu+λV(x)u=μup−1+u2∗−1,x∈RN.-\Delta u+\lambda V\left(x)u=\mu {u}^{p-1}+{u}^{{2}^{\ast }-1},\hspace{1.0em}x\in {{\mathbb{R}}}^{N}.By employing the Ljusternik-Schnirelmann category, for λ\lambda big enough and μ\mu small enough, the aforementioned problem has at last cat(Ω){\rm{cat}}\left(\Omega )positive solutions.For more results about these kinds of problems and fractional Schrödinger equations, see, for example, [9,10, 11,12,13, 14,15,16, 17,18,19, 20,21,22, 23,24] and references therein. Motivated by the aforementioned results, we consider equation (1.1). The potential function V(x)V\left(x)satisfies (V1)V(x)∈C(RN,R)V\left(x)\in C\left({{\mathbb{R}}}^{N},{\mathbb{R}})such that V(x)≥0V\left(x)\ge 0, Ω≔intV−1(0)\Omega := {\rm{int}}{V}^{-1}\left(0)is a nonempty open set of class C0.1{C}^{0.1}with bounded boundary and V−1(0)=Ω¯{V}^{-1}\left(0)=\bar{\Omega };(V2)There exists M0>0{M}_{0}\gt 0such that μ({x∈RN:V(x)≤M0})<∞,\mu (\left\{x\in {{\mathbb{R}}}^{N}:V\left(x)\le {M}_{0}\right\})\lt \infty ,where μ\mu denotes the Lebesgue measure on RN{{\mathbb{R}}}^{N}.We first introdcue some notations. For s∈(0,1),p∈[1,+∞)s\in \left(0,1),p\in {[}1,+\infty ), define Ws,p(RN)≔u∈Lp(RN):∣u(x)−u(y)∣∣x−y∣Np+s∈Lp(RN×RN),\hspace{-39.85em}{W}^{s,p}\left({{\mathbb{R}}}^{N}):= \left\{\phantom{\rule[-1.25em]{}{0ex}},u\in {L}^{p}\left({{\mathbb{R}}}^{N}):\frac{| u\left(x)-u(y)| }{| x-y\hspace{-0.25em}{| }^{\tfrac{N}{p}+s}}\in {L}^{p}\left({{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N})\right\},endowed with the norm ‖u‖Ws,p(RN)≔∫RN∣u∣pdx+∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+spdxdy1p,\hspace{-40.35em}\Vert u{\Vert }_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}:= {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{p}{\rm{d}}x+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+sp}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}},where the term [u]Ws,p(RN)≔∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy1p\hspace{-40.1em}{\left[u]}_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}:= {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}is the so-called Gagliardo (semi)norm of uu. Moreover, we define Ws,p(Ω)≔u∈Lp(Ω):∣u(x)−u(y)∣∣x−y∣Np+s∈Lp(Ω×Ω),\hspace{-38.75em}{W}^{s,p}\left(\Omega ):= \left\{u\in {L}^{p}\left(\Omega ):\frac{| u\left(x)-u(y)| }{| x-y\hspace{-0.25em}{| }^{\tfrac{N}{p}+s}}\in {L}^{p}\left(\Omega \times \Omega )\right\},endowed with the norm ‖u‖Ws,p(Ω)≔∫Ω∣u∣pdx+∫Ω∫Ω∣u(x)−u(y)∣p∣x−y∣N+spdxdy1p.\hspace{-39.65em}\Vert u{\Vert }_{{W}^{s,p}\left(\Omega )}:= {\left(\mathop{\int }\limits_{\Omega }| u{| }^{p}{\rm{d}}x+\mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+sp}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}.Let Eλ≔u∈Ws,p(RN):∫RNλV(x)∣u(x)∣pdx<∞,\hspace{-33.8em}{E}_{\lambda }:= \left\{u\in {W}^{s,p}\left({{\mathbb{R}}}^{N}):\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\lt \infty \right\},with the norm ‖u‖λ=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣u(x)∣pdx1p.\hspace{-35.5em}\Vert u{\Vert }_{\lambda }={\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{1}{p}}.The energy functional associated with (1.1) is (1.2)Jλ(u)=1p∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λp∫RNV(x)∣u(x)∣pdx−1q∫RNu+(x)qdxforu∈Eλ,{J}_{\lambda }\left(u)=\frac{1}{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{\lambda }{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\frac{1}{q}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}u\in {E}_{\lambda },where u+=max{u,0}{u}^{+}=\max \left\{u,0\right\}. Then, we can define the Nehari manifold ℳλ≔u∈Eλ⧹{0}:∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx=∫RNu+(x)qdx{{\mathcal{ {\mathcal M} }}}_{\lambda }:= \left\{u\in {E}_{\lambda }\setminus \left\{0\right\}:\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x=\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\right\}and cλ≔inf{Jλ(u):u∈ℳλ}.{c}_{\lambda }:= \inf \left\{{J}_{\lambda }\left(u):u\in {{\mathcal{ {\mathcal M} }}}_{\lambda }\right\}.Consider the following “limit” problem of (1.1) (1.3)(−Δ)psu(x)=u(x)q−1,x∈Ω,u(x)≥0,x∈Ω,u(x)=0,x∈RN⧹Ω,\left\{\begin{array}{l}{\left(-\Delta )}_{p}^{s}u\left(x)=u{\left(x)}^{q-1},\hspace{1em}x\in \Omega ,\\ u\left(x)\ge 0,\hspace{1em}x\in \Omega ,\\ u\left(x)=0,\hspace{1em}x\in {{\mathbb{R}}}^{N}\setminus \Omega ,\end{array}\right.Define a subspace E0{E}_{0}of Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})as follows: (1.4)E0≔{u∈Ws,p(RN):u(x)=0inRN⧹Ω}{E}_{0}:= \{u\in {W}^{s,p}\left({{\mathbb{R}}}^{N}):u\left(x)=0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega \}trΩE0={u∣Ω:u∈E0}.\hspace{-19.55em}{{\rm{tr}}}_{\Omega }{E}_{0}=\{u{| }_{\Omega }:u\in {E}_{0}\}.The energy functional associated with (1.3) can be defined by Φ(u)=1p∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy−1q∫Ωu+(x)qdxforu∈E0.\Phi \left(u)=\frac{1}{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y-\frac{1}{q}\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}u\in {E}_{0}.Then, the associated Nehari manifold is N≔u∈E0⧹{0}:∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy=∫Ωu+(x)qdx{\mathcal{N}}:= \left\{u\in {E}_{0}\setminus \left\{0\right\}:\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y=\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x\right\}and c(Ω)≔inf{Φ(u):u∈N}.c\left(\Omega ):= \inf \left\{\Phi \left(u):u\in {\mathcal{N}}\right\}.Definition 1.1A function uλ(x){u}_{\lambda }\left(x)is a ground state solution of (1.1) if cλ{c}_{\lambda }is achieved by uλ∈ℳλ{u}_{\lambda }\in {{\mathcal{ {\mathcal M} }}}_{\lambda }, which is a critical point of Jλ{J}_{\lambda }. Similarly, a function u(x)u\left(x)is a ground state solution of (1.3) if c(Ω)c\left(\Omega )is achieved by u∈Nu\in {\mathcal{N}}, which is a critical point of Φ\Phi .Definition 1.2Let XXbe a Banach space, φ∈C1(X,R).\varphi \in {C}^{1}\left(X,{\mathbb{R}}).The function φ\varphi satisfies the (PS)c{\left(PS)}_{c}condition if any sequence {un}⊆X\left\{{u}_{n}\right\}\subseteq X, such that (1.5)φ(un)→c,φ′(un)→0\varphi \left({u}_{n})\to c,\varphi ^{\prime} \left({u}_{n})\to 0has a convergent subsequence. The sequence {un}\left\{{u}_{n}\right\}that satisfies (1.5) is called to be a (PS)c{\left(PS)}_{c}sequence of φ\varphi .Our main results read as follows:Theorem 1.3Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold, then for λ\lambda large, (1.1) has a ground state solution uλ(x){u}_{\lambda }\left(x). Furthermore, any sequence λn→∞{\lambda }_{n}\to \infty , {uλn(x)}\left\{{u}_{{\lambda }_{n}}\left(x)\right\}has a subsequence such that uλn{u}_{{\lambda }_{n}}converges in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})along the subsequence to a ground state solution uuof (1.3).Theorem 1.4Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold. Let un,n∈N{u}_{n},n\in {\mathbb{N}}be a sequence of solutions of (1.1) with λ\lambda being replaced by λn{\lambda }_{n}(λn→∞{\lambda }_{n}\to \infty as n→∞n\to \infty ) such that lim supn→∞Jλ(un)<∞{\mathrm{lim\; sup}}_{n\to \infty }{J}_{\lambda }\left({u}_{n})\lt \infty . Then, un(x){u}_{n}\left(x)converges strongly along a subsequence in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})to a solution uuof (1.3).The following paper is organized as follows: In Section 2, we will give some preliminary results. Section 3 is devoted to the “limit” problem, and Section 4 contains the proofs of the main results. CCdenotes various generic positive constants, and o(1)o\left(1)will be used to represent quantities that tend to 0 as λ(orn)→∞\lambda \left(\hspace{0.1em}\text{or}\hspace{0.1em}\hspace{0.33em}n)\to \infty .2Preliminary resultsLemma 2.1Let λ0>0{\lambda }_{0}\gt 0be a fixed constant. Then, for λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, V(x)V\left(x)satisfying (V1)\left({V}_{1})and (V2)\left({V}_{2}), Eλ{E}_{\lambda }is continuously embedded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})uniformly in λ\lambda .ProofBy the definition of Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and Eλ{E}_{\lambda }, we only need to prove the following inequality: (2.1)∫RN∣u(x)∣pdx≤C∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣u(x)∣pdx.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u\left(x){| }^{p}{\rm{d}}x\le C\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\right).Define D≔{x∈RN:V(x)≤M0}\hspace{-12.4em}D:= \{x\in {{\mathbb{R}}}^{N}:V\left(x)\le {M}_{0}\}and Dδ0≔{x∈RN:dist(x,D)≤δ0}.{D}^{{\delta }_{0}}:= \{x\in {{\mathbb{R}}}^{N}:{\rm{dist}}\left(x,D)\le {\delta }_{0}\}.Take ζ∈C∞(RN,R)\zeta \in {C}^{\infty }\left({{\mathbb{R}}}^{N},{\mathbb{R}}), 0≤ζ≤10\le \zeta \le 1, satisfying (2.2)ζ(x)=1,x∈D,0,x∉Dδ0,∣∇ζ∣≤C/δ0.\hspace{-14.6em}\zeta \left(x)=\left\{\begin{array}{ll}1,& x\in D,\\ 0,& x\notin {D}^{{\delta }_{0}},\end{array}\right.\hspace{1.0em}| \nabla \zeta | \le C\hspace{-0.08em}\text{/}\hspace{-0.08em}{\delta }_{0}.Then, for any function u∈Eλu\in {E}_{\lambda }, we can obtain (2.3)∫RN(1−ζp)∣u(x)∣pdx=∫RN⧹D(1−ζp)∣u(x)∣pdx+∫D(1−ζp)∣u(x)∣pdx≤1λ0M0λ∫RNV(x)∣u(x)∣pdx\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x=\mathop{\int }\limits_{{{\mathbb{R}}}^{N}\setminus D}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x+\mathop{\int }\limits_{D}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x\le \frac{1}{{\lambda }_{0}{M}_{0}}\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}xand (2.4)∫RNζp∣u(x)∣pdx=∫Dδ0ζp∣u(x)∣pdx≤μ(Dδ0)1−pps∗∫Dδ0∣u(x)∣ps∗dxpps∗≤Cμ(Dδ0)1−pps∗∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy,\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\zeta }^{p}| u\left(x){| }^{p}{\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{{D}^{{\delta }_{0}}}{\zeta }^{p}| u\left(x){| }^{p}{\rm{d}}x\\ & \le & \mu {\left({D}^{{\delta }_{0}})}^{1-\tfrac{p}{{p}_{s}^{\ast }}}{\left(\mathop{\displaystyle \int }\limits_{{D}^{{\delta }_{0}}}| u\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{p}{{p}_{s}^{\ast }}}\\ & \le & C\mu {\left({D}^{{\delta }_{0}})}^{1-\tfrac{p}{{p}_{s}^{\ast }}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y,\end{array}where we have used (V2)\left({V}_{2})and the Sobolev trace inequality ∫RN∣u(x)∣ps∗dx1/ps∗≤C∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy1/p,{\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{1\text{/}{p}_{s}^{\ast }}\le C{\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{1\text{/}p},for u∈Ws,p(RN)u\in {W}^{s,p}\left({{\mathbb{R}}}^{N})and C=C(N,p,s)>0C=C\left(N,p,s)\gt 0. Thus, (2.1) follows from (2.3) and (2.4).□Lemma 2.2There exists σ>0\sigma \gt 0independent of λ\lambda , such that ‖u‖λ≥σ\Vert u{\Vert }_{\lambda }\ge \sigma for all u∈ℳλu\in {{\mathcal{ {\mathcal M} }}}_{\lambda }.ProofFrom Lemma 2.1, for any u∈ℳλu\in {{\mathcal{ {\mathcal M} }}}_{\lambda }, 0=⟨Jλ′(u),u⟩=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx−∫RNu+(x)qdx≥‖u‖λp−C‖u‖Ws,p(RN)q≥‖u‖λp−C‖u‖λq,\begin{array}{rcl}0=\langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle & =& \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\\ & \ge & \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}^{q}\\ & \ge & \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{\lambda }^{q},\end{array}where C>0C\gt 0is independent of λ≥0\lambda \ge 0. The aforementioned inequality implies that ‖u‖λq−p≥1C\Vert u{\Vert }_{\lambda }^{q-p}\ge \frac{1}{C}. Choosing σ=1C1q−p\sigma ={\left(\frac{1}{C}\right)}^{\tfrac{1}{q-p}}, we obtain ‖u‖λ≥σ\Vert u{\Vert }_{\lambda }\ge \sigma .□Lemma 2.3Let λ0{\lambda }_{0}be a fixed positive constant, there exists c0>0{c}_{0}\gt 0independent of λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, such that if {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }, then either c≥c0c\ge {c}_{0}or c=0c=0. Moreover, (2.5)lim supn→∞‖un‖λp≤pqq−pc.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \frac{pq}{q-p}c.ProofFrom the definition of (PS)c{\left(PS)}_{c}sequence, c+‖un‖λ⋅o(1)=Jλ(un)−1q⟨Jλ′(un),un⟩=1p−1q∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣un(x)∣pdx=q−ppq‖un‖λp.\begin{array}{rcl}c+\Vert {u}_{n}{\Vert }_{\lambda }\cdot o\left(1)& =& {J}_{\lambda }\left({u}_{n})-\frac{1}{q}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle \\ & =& \left(\frac{1}{p}-\frac{1}{q}\right)\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & =& \frac{q-p}{pq}\Vert {u}_{n}{\Vert }_{\lambda }^{p}.\end{array}Then, (2.5) holds. On the other side, there is a constant C>0C\gt 0independent of λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, such that ⟨Jλ′(u),u⟩=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx−∫RNu+(x)qdx≥‖u‖λp−C‖u‖λq.\langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle =\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\ge \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{\lambda }^{q}.Thus, there exists σ1>0{\sigma }_{1}\gt 0independent of λ\lambda , such that (2.6)14‖u‖λp≤⟨Jλ′(u),u⟩for‖u‖λ<σ1.\frac{1}{4}\Vert u{\Vert }_{\lambda }^{p}\le \langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle \hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}\Vert u{\Vert }_{\lambda }\lt {\sigma }_{1}.If c<σ1p(q−p)pqc\lt \frac{{\sigma }_{1}^{p}\left(q-p)}{pq}, then lim supn→∞‖un‖λp≤cpqq−p<σ1p.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \frac{cpq}{q-p}\lt {\sigma }_{1}^{p}.Hence, ‖un‖λ<σ1\Vert {u}_{n}{\Vert }_{\lambda }\lt {\sigma }_{1}for nnlarge. It follows from (2.6) that 14‖un‖λp≤⟨Jλ′(un),un⟩=o(1)‖un‖λ,\frac{1}{4}\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle =o\left(1)\Vert {u}_{n}{\Vert }_{\lambda },which implies ‖un‖λ→0\Vert {u}_{n}{\Vert }_{\lambda }\to 0as n→∞n\to \infty . Therefore, Jλ(un)→0{J}_{\lambda }\left({u}_{n})\to 0, that is, c=0c=0. Thus, c0=σ1p(q−p)qp{c}_{0}=\frac{{\sigma }_{1}^{p}\left(q-p)}{qp}is as required.□Lemma 2.4There exists δ0>0{\delta }_{0}\gt 0, such that any (PS)c{\left(PS)}_{c}sequence {un}\left\{{u}_{n}\right\}of Jλ{J}_{\lambda }with λ≥0\lambda \ge 0and c>0c\gt 0satisfies(2.7)lim infn→∞‖un+‖Lq(RN)q≥δ0c.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\Vert {u}_{n}^{+}{\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q}\ge {\delta }_{0}c.ProofFrom the definition of (PS)c{\left(PS)}_{c}sequence, c=limn→∞Jλ(un)−1p⟨Jλ′(un),un⟩=1p−1qlimn→∞∫RNun+(x)qdx=(q−p)qplimn→∞‖un+(x)‖Lq(RN)q,c=\mathop{\mathrm{lim}}\limits_{n\to \infty }\left({J}_{\lambda }\left({u}_{n})-\frac{1}{p}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle \right)=\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x=\frac{\left(q-p)}{qp}\mathop{\mathrm{lim}}\limits_{n\to \infty }\Vert {u}_{n}^{+}\left(x){\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q},which implies (2.7) with δ0≤qpq−p{\delta }_{0}\le \frac{qp}{q-p}.□Lemma 2.5Let C1{C}_{1}be any fixed constant. Then, for any ε>0\varepsilon \gt 0, there exists Λε>0{\Lambda }_{\varepsilon }\gt 0and Rε>0{R}_{\varepsilon }\gt 0, such that if {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }with λ≥Λε,c≤C1\lambda \ge {\Lambda }_{\varepsilon },c\le {C}_{1}, then(2.8)lim supn→∞∫BRεcun+(x)qdx≤ε,\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{{R}_{\varepsilon }}^{c}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x\le \varepsilon ,where BRεc={x∈RN:∣x∣≥Rε}{B}_{{R}_{\varepsilon }}^{c}=\left\{x\in {{\mathbb{R}}}^{N}:| x| \ge {R}_{\varepsilon }\right\}.ProofFor R>0R\gt 0, let A(R)≔{x∈RN:∣x∣>R,V(x)≥M0}A\left(R):= \left\{x\in {{\mathbb{R}}}^{N}:| x| \gt R,V\left(x)\ge {M}_{0}\right\}and B(R)≔{x∈RN:∣x∣>R,V(x)<M0}.\hspace{0.3em}B\left(R):= \left\{x\in {{\mathbb{R}}}^{N}:| x| \gt R,V\left(x)\lt {M}_{0}\right\}.It follows from Lemma 2.3 that (2.9)∫A(R)∣un(x)∣pdx≤1λM0∫RNλV(x)∣un(x)∣pdx≤1λM0∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣un(x)∣pdx≤1λM0pqq−pC1+o(1).\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{A\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x& \le & \frac{1}{\lambda {M}_{0}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & \le & \frac{1}{\lambda {M}_{0}}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & \le & \frac{1}{\lambda {M}_{0}}\left(\frac{pq}{q-p}{C}_{1}+o\left(1)\right).\end{array}From Hölder inequality and (2.5), we can see that, for 1<r<N/(N−ps)1\lt r\lt N\hspace{-0.08em}\text{/}\hspace{-0.01em}\left(N-ps), (2.10)∫B(R)∣un(x)∣pdx≤∫RN∣un(x)∣prdx1/rμ(B(R))1/r′≤C‖un‖λp⋅μ(B(R))1/r′≤Cpqq−pC0⋅μ(B(R))1/r′,\mathop{\int }\limits_{B\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\le {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {u}_{n}\left(x){| }^{pr}{\rm{d}}x\right)}^{1\text{/}r}\mu {(B\left(R))}^{1\text{/}r^{\prime} }\le C\Vert {u}_{n}{\Vert }_{\lambda }^{p}\cdot \mu {\left(B\left(R))}^{1\text{/}r^{\prime} }\le C\frac{pq}{q-p}{C}_{0}\cdot \mu {\left(B\left(R))}^{1\text{/}r^{\prime} },where C=C(N,r)>0C=C\left(N,r)\gt 0and 1/r+1/r′=11\hspace{0.1em}\text{/}r+1\text{/}\hspace{0.1em}r^{\prime} =1. By interpolation inequality and Sobolev embedding inequality, we can obtain ∫BRcun+(x)qdx≤∫BRc∣un(x)∣pdxq(1−θ)p⋅∫BRc∣un(x)∣ps∗dxqθps∗≤∫BRc∣un(x)∣pdxq(1−θ)p∫RN∣un(x)∣ps∗dxqθps∗≤C∫BRc∣un(x)∣pdxq(1−θ)p∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdyqθp≤C∫A(R)∣un(x)∣pdx+∫B(R)∣un(x)∣pdxq(1−θ)p‖un‖λqθ,\hspace{-28.55em}\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x& \le & {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}\cdot {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{q\theta }{{p}_{s}^{\ast }}}\\ & \le & {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}{\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| {u}_{n}\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{q\theta }{{p}_{s}^{\ast }}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}{\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{q\theta }{p}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{A\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x+\mathop{\displaystyle \int }\limits_{B\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}\Vert {u}_{n}{\Vert }_{\lambda }^{q\theta },\end{array}where θ=Nsq−ppq\theta =\frac{N}{s}\frac{q-p}{pq}. Then, the result follows from (2.9), (2.10) and (V2)\left({V}_{2}).□Lemma 2.6(Brézis-Lieb lemma, 1983) Let {un}⊂Lp(RN)\left\{{u}_{n}\right\}\subset {L}^{p}\left({{\mathbb{R}}}^{N}), 1≤p<∞1\le p\lt \infty . If(a){un}\left\{{u}_{n}\right\}is bounded in Lp(RN){L}^{p}\left({{\mathbb{R}}}^{N}),(b)un→u{u}_{n}\to ualmost everywhere on RN{{\mathbb{R}}}^{N}, then (2.11)limn→∞(∣un∣pp−∣un−u∣pp)=∣u∣pp.\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(| {u}_{n}\hspace{-0.25em}{| }_{p}^{p}-| {u}_{n}-u\hspace{-0.25em}{| }_{p}^{p})=| u\hspace{-0.25em}{| }_{p}^{p}.Lemma 2.7Let λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0be fixed and let {un}\left\{{u}_{n}\right\}be a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }. Then, up to a subsequence, un⇀u{u}_{n}\rightharpoonup uin Eλ{E}_{\lambda }with uubeing a weak solution of (1.1). Moreover, un1=un−u{u}_{n}^{1}={u}_{n}-uis (PS)c′{\left(PS)}_{c^{\prime} }sequence with c′=c−Jλ(u)c^{\prime} =c-{J}_{\lambda }\left(u).ProofBy Lemma 2.3, {un}\left\{{u}_{n}\right\}is bounded in Eλ{E}_{\lambda }. Then, up to a subsequence un⇀u{u}_{n}\rightharpoonup uin Eλ{E}_{\lambda }as n→∞n\to \infty , and (2.12)un⇀uinWs,p(RN),\hspace{-15.95em}{u}_{n}\rightharpoonup u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{W}^{s,p}\left({{\mathbb{R}}}^{N}),(2.13)un⇀uinLq(RN),p≤q<ps∗,\hspace{-15.95em}{u}_{n}\rightharpoonup u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}^{q}\left({{\mathbb{R}}}^{N}),\hspace{1em}p\le q\lt {p}_{s}^{\ast },(2.14)un→uinLlocq(RN),p≤q<ps∗,{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}_{{\rm{loc}}}^{q}\left({{\mathbb{R}}}^{N}),\hspace{1em}p\le q\lt {p}_{s}^{\ast },(2.15)un→ua.e. inRN,{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{a.e. in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{N},where ps∗=NpN−ps{p}_{s}^{\ast }=\frac{Np}{N-ps}is the fractional critical Sobolev exponent. Hence, for any φ∈Eλ\varphi \in {E}_{\lambda }, we have ⟨Jλ′(un),φ⟩=∫RN∫RN∣un(x)−un(y)∣p−2(un(x)−un(y))(φ(x)−φ(y))∣x−y∣N+psdxdy+λ∫RNV(x)∣un(x)∣p−2un(x)φ(x)dx−∫RNun+(x)q−1φ(x)dx→∫RN∫RN∣u(x)−u(y)∣p−2(u(x)−u(y))(φ(x)−φ(y))∣x−y∣N+psdxdy+λ∫RNV(x)u(x)p−1φ(x)dx−∫RNu+(x)q−1φ(x)dx=⟨Jλ′(u),φ⟩.\begin{array}{rcl}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle & =& \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y)\hspace{-0.25em}{| }^{p-2}\left({u}_{n}\left(x)-{u}_{n}(y))\left(\varphi \left(x)-\varphi (y))}{| x-y\hspace{-0.25em}\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p-2}{u}_{n}\left(x)\varphi \left(x){\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q-1}\varphi \left(x){\rm{d}}x\\ & \to & \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p-2}\left(u\left(x)-u(y))\left(\varphi \left(x)-\varphi (y))}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)u{\left(x)}^{p-1}\varphi \left(x){\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q-1}\varphi \left(x){\rm{d}}x=\langle {J}_{\lambda }^{^{\prime} }\left(u),\varphi \rangle .\end{array}Therefore, (2.16)⟨Jλ′(u),φ⟩=limn→∞⟨Jλ′(un),φ⟩=0,\langle {J}_{\lambda }^{^{\prime} }\left(u),\varphi \rangle =\mathop{\mathrm{lim}}\limits_{n\to \infty }\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle =0,which implies that uuis a critical point of Jλ{J}_{\lambda }.Let un1=un−u{u}_{n}^{1}={u}_{n}-u, we will show that as n→∞n\to \infty , (2.17)Jλ(un1)→c−Jλ(u){J}_{\lambda }\left({u}_{n}^{1})\to c-{J}_{\lambda }\left(u)and (2.18)Jλ′(un1)→0.{J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1})\to 0.To show (2.17), we observe that (2.19)Jλ(un1)=1p∫RN∫RN∣un1(x)−un1(y)∣p∣x−y∣N+psdxdy+λp∫RNV(x)∣un1(x)∣pdx−1q∫RNun1+(x)qdx=Jλ(un)−Jλ(u)+λp∫RNV(x)(∣un1(x)∣p−∣un(x)∣p+∣u(x)∣p)dx+1p∫RN∫RN∣un1(x)−un1(y)∣p−∣un(x)−un(y)∣p+∣u(x)−u(y)∣p∣x−y∣N+psdxdy+1q∫RNun+(x)qdx−1q∫RNun1+(x)qdx−1q∫RNu+(x)qdx.\begin{array}{rcl}{J}_{\lambda }\left({u}_{n}^{1})& =& \frac{1}{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}^{1}\left(x)-{u}_{n}^{1}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{\lambda }{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}^{1}\left(x){| }^{p}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x\\ & =& {J}_{\lambda }\left({u}_{n})-{J}_{\lambda }\left(u)+\frac{\lambda }{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)\left(| {u}_{n}^{1}\left(x){| }^{p}-| {u}_{n}\left(x){| }^{p}+| u\left(x){| }^{p}){\rm{d}}x\\ & & +\frac{1}{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}^{1}\left(x)-{u}_{n}^{1}(y){| }^{p}-| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}+| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x.\end{array}From Lemma 2.6, ∫RNun+(x)qdx−∫RNu+(x)qdx−∫RNun1+(x)qdx→0{\int }_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-{\int }_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x-{\int }_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x\to 0as n→∞n\to \infty . Conversely, we know that ‖un‖λp−‖u‖λp−‖un1‖λp→0\Vert {u}_{n}{\Vert }_{\lambda }^{p}-\Vert u{\Vert }_{\lambda }^{p}-\Vert {u}_{n}^{1}{\Vert }_{\lambda }^{p}\to 0, as n→∞n\to \infty . Thus, from (2.19), we indeed have obtained (2.17). Now we come to show (2.18). From (2.16), we have for any φ∈Eλ\varphi \in {E}_{\lambda }⟨Jλ′(un1),φ⟩=⟨Jλ′(un),φ⟩−∫RN(un1+)q−1φ(x)dx+∫RN(un+)q−1φ(x)dx−∫RN(u+)q−1φ(x)dx+o(1).\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1}),\varphi \rangle =\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle -\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}_{n}^{1+})}^{q-1}\varphi \left(x){\rm{d}}x+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}_{n}^{+})}^{q-1}\varphi \left(x){\rm{d}}x-\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}^{+})}^{q-1}\varphi \left(x){\rm{d}}x+o\left(1).Since Jλ′(un)→0{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to 0and un⇀u{u}_{n}\rightharpoonup uin Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N}), we have limn→∞sup‖φ‖λ≤1∫RN((un1+)q−1(x)φ(x)−(un+)q−1φ(x)+(u+)q−1φ(x))dx=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\Vert \varphi {\Vert }_{\lambda }\le 1}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}({\left({u}_{n}^{1+})}^{q-1}\left(x)\varphi \left(x)-{\left({u}_{n}^{+})}^{q-1}\varphi \left(x)+{\left({u}^{+})}^{q-1}\varphi \left(x)){\rm{d}}x=0.Thus, we have limn→∞⟨Jλ′(un1),φ⟩=0for anyφ∈Eλ,\mathop{\mathrm{lim}}\limits_{n\to \infty }\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1}),\varphi \rangle =0\hspace{0.33em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\varphi \in {E}_{\lambda },which implies (2.18), and this completes the proof.□Proposition 2.8Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold. Then, for any C0>0{C}_{0}\gt 0, there exists Λ0>0{\Lambda }_{0}\gt 0such that Jλ{J}_{\lambda }satisfies the (PS)c{\left(PS)}_{c}condition for all λ≥Λ0\lambda \ge {\Lambda }_{0}and c≤C0c\le {C}_{0}.ProofChoose 0<ε<δ0c0/20\lt \varepsilon \lt {\delta }_{0}{c}_{0}\hspace{-0.08em}\text{/}\hspace{-0.08em}2, where c0{c}_{0}and δ0{\delta }_{0}are the constants in Lemmas 2.3 and 2.4, respectively. Let Λ0≔Λε{\Lambda }_{0}:= {\Lambda }_{\varepsilon }, where Λε>0{\Lambda }_{\varepsilon }\gt 0is from Lemma 2.5.Assume {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }with λ≥Λ0\lambda \ge {\Lambda }_{0}and c≤C0c\le {C}_{0}. By Lemma 2.7, un1=un−u{u}_{n}^{1}={u}_{n}-uis a (PS)c′{\left(PS)}_{c^{\prime} }sequence of Jλ{J}_{\lambda }with c′=c−Jλ(u)c^{\prime} =c-{J}_{\lambda }\left(u). If c′>0c^{\prime} \gt 0, it follows from Lemma 2.3 that c′≥c0c^{\prime} \ge {c}_{0}. From Lemma 2.4, we can obtain lim infn→∞‖un1+(⋅)‖Lq(RN)q≥δ0c′≥δ0c0.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\Vert {u}_{n}^{1+}\left(\cdot ){\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q}\ge {\delta }_{0}c^{\prime} \ge {\delta }_{0}{c}_{0}.Conversely, Lemma 2.5 implies lim supn→∞∫BRεcun1+(x)q≤ε<δ0c02.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{{R}_{\varepsilon }}^{c}}{u}_{n}^{1+}{\left(x)}^{q}\le \varepsilon \lt \frac{{\delta }_{0}{c}_{0}}{2}.Noting un1→0{u}_{n}^{1}\to 0in Llocq(RN){L}_{{\rm{loc}}}^{q}\left({{\mathbb{R}}}^{N}), p≤q<ps∗p\le q\lt {p}_{s}^{\ast }, a contradiction follows from the aforementioned two inequalities. Therefore, c′=0c^{\prime} =0. Thus, un1→0{u}_{n}^{1}\to 0in Eλ{E}_{\lambda }by Lemma 2.3.□Corollary 2.9For any q∈(p,ps∗)q\in \left(p,{p}_{s}^{\ast }), there exists Λ0>0{\Lambda }_{0}\gt 0, such that cλ{c}_{\lambda }is achieved for all λ≥Λ0\lambda \ge {\Lambda }_{0}at some uλ∈Eλ{u}_{\lambda }\in {E}_{\lambda }, which is a ground state solution of (1.1).ProofBy Ekeland variational principle, there is a PS sequence un∈Eλ{u}_{n}\in {E}_{\lambda }, such that Jλ(un)→cλandJλ′(un)→0.{J}_{\lambda }\left({u}_{n})\to {c}_{\lambda }\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to 0.By Proposition 2.8, there exists some uλ∈Eλ{u}_{\lambda }\in {E}_{\lambda }, such that, up to subsequence, un→uλ{u}_{n}\to {u}_{\lambda }in Eλ{E}_{\lambda }as n→∞n\to \infty and λ\lambda is sufficiently large. It is not difficult to show that Jλ(un)→Jλ(uλ)andJλ′(un)→Jλ′(uλ).{J}_{\lambda }\left({u}_{n})\to {J}_{\lambda }\left({u}_{\lambda })\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to {J}_{\lambda }^{^{\prime} }\left({u}_{\lambda }).Therefore, we have Jλ(uλ)=cλandJλ′(uλ)=0.{J}_{\lambda }\left({u}_{\lambda })={c}_{\lambda }\hspace{0.25em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.25em}{J}_{\lambda }^{^{\prime} }\left({u}_{\lambda })=0.This means that uλ{u}_{\lambda }is a ground state solution of (1.1).□3Limit problemLemma 3.1Let 1<p<q<ps∗≔pNN−ps1\lt p\lt q\lt {p}_{s}^{\ast }:= \frac{pN}{N-ps}, N≥2N\ge 2. Then, trΩE0t{r}_{\Omega }{E}_{0}is compactly embedded in Lq(Ω){L}^{q}\left(\Omega ).ProofSince trΩE0⊂Ws,p(Ω)t{r}_{\Omega }{E}_{0}\subset {W}^{s,p}\left(\Omega )and Ws,p(Ω)↪Lp(Ω){W}^{s,p}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{p}\left(\Omega )are compact for p<q<ps∗p\lt q\lt {p}_{s}^{\ast }, N≥2N\ge 2, the result follows.□Lemma 3.2The infimum c(Ω)c\left(\Omega )is achieved by a function u∈Nu\in {\mathcal{N}}, which is a ground state solution of (1.3).ProofBy Ekeland variational principle, there is a PS sequence un∈E0{u}_{n}\in {E}_{0}, such that Φ(un)→c(Ω)andΦ′(un)→0.\Phi \left({u}_{n})\to c\left(\Omega )\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\Phi ^{\prime} \left({u}_{n})\to 0.Thus, by Lemma 3.1, we can easily obtain a subsequence of {un}\left\{{u}_{n}\right\}(still denote it itself), such that un→u{u}_{n}\to uin E0{E}_{0}. Therefore, uuis a ground state solution of (1.3).□Remark 3.3Assume set Ω=intV−1(0)\Omega ={\rm{int}}{V}^{-1}\left(0)has more than one isolated component, for example, Ω=Ω1∪Ω2\Omega ={\Omega }_{1}\cup {\Omega }_{2}with Ω1∩Ω2=∅{\Omega }_{1}\cap {\Omega }_{2}=\varnothing . Suppose that u∈Nu\in {\mathcal{N}}is a nonnegative solution of (1.3) with u(x)=0u\left(x)=0in Ω1{\Omega }_{1}and u(x)≩0u\left(x)\gneqq 0in Ω2{\Omega }_{2}. Then, we have (−Δ)psu(x)=∫RN∣u(x)−u(y)∣p−2(u(x)−u(y))∣x−y∣N+psdy<0{\left(-\Delta )}_{p}^{s}u\left(x)={\int }_{{{\mathbb{R}}}^{N}}\frac{| \hspace{-0.25em}u\left(x)-u(y)\hspace{-0.25em}{| }^{p-2}\left(u\left(x)-u(y))}{| \hspace{-0.25em}x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}y\lt 0in Ω1{\Omega }_{1}. Conversely, (−Δ)psu(x)=u(x)q−1=0{\left(-\Delta )}_{p}^{s}u\left(x)=u{\left(x)}^{q-1}=0for x∈Ω1x\in {\Omega }_{1}. This contradiction shows that the nonnegative solution u(x)u\left(x)of (1.3) must be u(x)≩0u\left(x)\gneqq 0in both Ω1{\Omega }_{1}and Ω2.{\Omega }_{2}.However, the Laplacian case can have a nonnegative solution u(x)u\left(x)satisfying u(x)=0u\left(x)=0in Ω1{\Omega }_{1}and u(x)≩0u\left(x)\gneqq 0in Ω2{\Omega }_{2}. The difference between the two phenomena is attributed to the nonlocality of fractional operators and the locality of Laplacian operators.4The proof of the main resultsLemma 4.1cλ→c(Ω){c}_{\lambda }\to c\left(\Omega )as λ→∞\lambda \to \infty .ProofFrom the definition of cλ{c}_{\lambda }and c(Ω)c\left(\Omega ), we know that cλ≤c(Ω){c}_{\lambda }\le c\left(\Omega ), λ>0\lambda \gt 0. Furthermore, cλ{c}_{\lambda }is monotone increasing about the parameter λ>0.\lambda \gt 0.Then, there exists a constant kk, such that limn→∞cλn=k,\mathop{\mathrm{lim}}\limits_{n\to \infty }{c}_{{\lambda }_{n}}=k,where λn→∞{\lambda }_{n}\to \infty . It follows from Lemma 2.3 that k>0k\gt 0. By Corollary 2.9, for nnlarge enough, there exists a sequence un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}, such that Jλn′(un)=0{J}_{{\lambda }_{n}}^{^{\prime} }\left({u}_{n})=0and Jλn(un)=cλn.{J}_{{\lambda }_{n}}\left({u}_{n})={c}_{{\lambda }_{n}}.If k<c(Ω)k\lt c\left(\Omega ), it is easy to see that {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}); thus, we can assume that un⇀u{u}_{n}\rightharpoonup uin EEand (4.1)un(x)→u(x)inLlocθ(RN)forp≤θ<ps∗.{u}_{n}\left(x)\to u\left(x)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})\hspace{1em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}p\le \theta \lt {p}_{s}^{\ast }.Claim 1: u∣Ωc=0u{| }_{{\Omega }^{c}}=0. In fact, if u∣Ωc≠0u\hspace{-0.25em}{| }_{{\Omega }^{c}}\ne 0, then there exists a compact subset F⊂ΩcF\subset {\Omega }^{c}with dist(F,Ω)>0\left(F,\Omega )\gt 0, such that u∣F≠0u{| }_{F}\ne 0. It follows from (4.1) that ∫F∣un(x)∣pdx→∫F∣u(x)∣pdx>0.\mathop{\int }\limits_{F}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to \mathop{\int }\limits_{F}| u\left(x){| }^{p}{\rm{d}}x\gt 0.However, there exists ε0>0{\varepsilon }_{0}\gt 0, such that V(x)≥ε0>0V\left(x)\ge {\varepsilon }_{0}\gt 0, x∈Fx\in F. Thus, Jλn(un)≥q−ppqλn∫FV(x)∣un(x)∣pdx≥q−ppqλnε0∫F∣un(x)∣pdx→∞asn→∞,{J}_{{\lambda }_{n}}\left({u}_{n})\ge \frac{q-p}{pq}{\lambda }_{n}\mathop{\int }\limits_{F}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\ge \frac{q-p}{pq}{\lambda }_{n}{\varepsilon }_{0}\mathop{\int }\limits_{F}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to \infty \hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,which is a contradiction. Therefore, u∈E0u\in {E}_{0}.Claim 2: un→u{u}_{n}\to uin Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N})for p<q<ps∗p\lt q\lt {p}_{s}^{\ast }. Indeed, if not, then by the concentration-compactness lemma from the study by Loins [25], there exist δ>0,ρ>0\delta \gt 0,\rho \gt 0and xn∈RN{x}_{n}\in {{\mathbb{R}}}^{N}with ∣xn∣→∞| {x}_{n}| \to \infty , such that lim infn→∞∫Bρ(xn)∣un(x)−u(x)∣pdx≥δ>0.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\ge \delta \gt 0.Then, we have Jλn(un)=q−ppq∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+q−ppq∫RNλnV(x)∣un(x)∣pdx≥q−ppqλn∫Bρ(xn)∩{x:V(x)≥M0}V(x)∣un(x)∣pdx=q−ppqλn∫Bρ(xn)∩{x:V(x)≥M0}V(x)∣un(x)−u(x)∣pdx≥q−ppqλnM0∫Bρ(xn)∣un(x)−u(x)∣pdx−M0∫Bρ(xn)∩{x:V(x)≤M0}∣un(x)∣pdx≥q−ppqλnM0∫Bρ(xn)∣un(x)−u(x)∣pdx−o(1)→∞asn→∞,\begin{array}{rcl}{J}_{{\lambda }_{n}}\left({u}_{n})& =& \frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\ge {M}_{0}\right\}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & =& \frac{q-p}{pq}{\lambda }_{n}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\ge {M}_{0}\right\}}V\left(x)| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\left({M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x-{M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\le {M}_{0}\right\}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\left({M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x-o\left(1)\right)\to \infty \hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,\end{array}as a contradiction. So un→u{u}_{n}\to uin Lp(RN){L}^{p}\left({{\mathbb{R}}}^{N}). Therefore, it is easy to see that u≥0u\ge 0is a solution for problem (1.3). Furthermore, k=limn→∞cλn=limn→∞Jλn(un)=limn→∞1p−1q∫RNun+(x)qdx=1p−1q∫Ωu+(x)qdx,k=\mathop{\mathrm{lim}}\limits_{n\to \infty }{c}_{{\lambda }_{n}}=\mathop{\mathrm{lim}}\limits_{n\to \infty }{J}_{{\lambda }_{n}}\left({u}_{n})=\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x=\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x,which means u∈Nu\in {\mathcal{N}}, and then, k≥c(Ω)k\ge c\left(\Omega ), a contradiction. Hence, limλ→∞cλ=c(Ω){\mathrm{lim}}_{\lambda \to \infty }\hspace{0.25em}{c}_{\lambda }=c\left(\Omega ).□Proof of Theorem 1.3By Corollary 2.9, there exists un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}, such that Jλn(un)=cλn{J}_{{\lambda }_{n}}\left({u}_{n})={c}_{{\lambda }_{n}}(λn→∞asn→∞{\lambda }_{n}\to \infty \hspace{0.25em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.25em}n\to \infty ). It is easy to see that {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). Then, without loss of generality, un⇀u{u}_{n}\rightharpoonup uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and un→u{u}_{n}\to uin Llocθ(RN){L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗p\lt \theta \lt {p}_{s}^{\ast }.Now we prove that un→u{u}_{n}\to ustrongly in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and uuis a ground state solution of (1.3). First, as the proof of Lemma 4.1, u≥0u\ge 0is a solution of problem (1.3) and un+→u+{u}_{n}^{+}\to {u}^{+}strongly in Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N}).Now we claim that λn∫RNV(x)∣un(x)∣pdx→0{\lambda }_{n}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to 0and ∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy→∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\to \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y.Indeed, if either lim infn→∞λn∫RNV(x)∣un(x)∣pdx>0\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }{\lambda }_{n}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\gt 0or lim infn→∞∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy>∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\gt \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y.Thus, we have ∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy<∫Ωu+(x)qdx.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\lt \mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x.Therefore, there is α∈(0,1)\alpha \in \left(0,1), such that αu∈N\alpha u\in {\mathcal{N}}and c(Ω)≤Φ(αu)=q−ppq∫RN∫RN∣αu(x)−αu(y)∣p∣x−y∣N+psdxdy<q−ppq∫RN∫RN∣u(x)−u(y)∣p∣x−y∣n+psdxdy≤limn→∞q−ppq∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+∫RNλnV(x)∣un(x)∣pdx=limn→∞Jλn(un)=c(Ω),\begin{array}{rcl}c\left(\Omega )& \le & \Phi \left(\alpha u)=\frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| \alpha u\left(x)-\alpha u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & \lt & \frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y\\ & \le & \mathop{\mathrm{lim}}\limits_{n\to \infty }\frac{q-p}{pq}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & =& \mathop{\mathrm{lim}}\limits_{n\to \infty }{J}_{{\lambda }_{n}}\left({u}_{n})=c\left(\Omega ),\end{array}which is a contradiction. By now we complete the proof of Theorem 1.3.□Proof of Theorem 1.4Suppose {un}⊂Ws,p(RN)\left\{{u}_{n}\right\}\subset {W}^{s,p}\left({{\mathbb{R}}}^{N})is a solution of (1.1) with λ\lambda being replaced by λn{\lambda }_{n}(λn→∞{\lambda }_{n}\to \infty as n→∞n\to \infty ). It follows from lim supn→∞Jλ(un)<∞{\mathrm{lim\; sup}}_{n\to \infty }{J}_{\lambda }\left({u}_{n})\lt \infty that such a sequence {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). Suppose that un⇀u{u}_{n}\rightharpoonup uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and un→u{u}_{n}\to uin Llocθ(RN){L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗.p\lt \theta \lt {p}_{s}^{\ast }.Similar to the proof of Lemma 4.1, u∣Ωc=0u{| }_{{\Omega }^{c}}=0and u∈E0u\in {E}_{0}is solution of (1.3). Moreover, un→u{u}_{n}\to uin Lθ(RN){L}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗p\lt \theta \lt {p}_{s}^{\ast }. Noting un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}and u∈Nu\in {\mathcal{N}}, we can obtain ∫RN∫RN∣un(x)−un(y)−u(x)+u(y)∣p∣x−y∣n+psdxdy+∫RNλnV(x)∣un(x)−u(x)∣pdx=∫RN∫RN∣un(x)−un(y)∣p∣x−y∣n+psdxdy+∫RNλnV(x)∣un(x)∣pdx−∫RN∫RN∣u(x)−u(y)∣p∣x−y∣n+psdxdy−∫RNλnV(x)∣u(x)∣pdx+o(1)=∫RNun+(x)qdx−∫Ωu+(x)qdx+o(1)=o(1).\begin{array}{l}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y)-u\left(x)+u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| u\left(x){| }^{p}{\rm{d}}x+o\left(1)\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x+o\left(1)=o\left(1).\end{array}Thus, un→u{u}_{n}\to uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). This completes the proof of Theorem 1.4.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Open Mathematics , Volume 20 (1): 13 – Jan 1, 2022

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de Gruyter
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© 2022 Yongpeng Chen and Miaomiao Niu, published by De Gruyter
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2391-5455
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10.1515/math-2022-0006
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Abstract

1Introduction and main resultsIn this paper, we consider the following nonlinear Schrödinger equations involving the fractional p-Laplacian (1.1)(−Δ)psu(x)+λV(x)u(x)p−1=u(x)q−1,x∈RN,u(x)≥0,u(x)∈Ws,p(RN),\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\left(-\Delta )}_{p}^{s}u\left(x)+\lambda V\left(x)u{\left(x)}^{p-1}=u{\left(x)}^{q-1},\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\ge 0,\hspace{1em}\hspace{1em}u\left(x)\in {W}^{s,p}\left({{\mathbb{R}}}^{N}),\end{array}\right.where λ>0\lambda \gt 0is a parameter, 1<p<q<NpN−sp1\lt p\lt q\lt \frac{Np}{N-sp}, N≥2N\ge 2, and V(x)V\left(x)is a real continuous function on RN{{\mathbb{R}}}^{N}.We are interested in the existence of ground state solutions for λ\lambda big enough, and their asymptotical behavior as λ→∞.\lambda \to \infty .As far as we know, these kinds of problems were first put forward in [1] by Bartsch and Wang, where they studied the Schrödinger equations. Under suitable conditions imposed on the potential, the loss of compactness caused by the whole space RN{{\mathbb{R}}}^{N}can be recovered when parameter λ\lambda is big enough. Then, many authors began studying the problems with potential well. A lot of results have been obtained.Bartsch and Parnet [2] also considered the nonlinear Schrödinger equation: −Δu+(a0(x)+λa(x))u=f(x,u),x∈RN,u(x)→0,∣x∣→∞,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\hspace{-0.25em}\Delta u+\left({a}_{0}\left(x)+\lambda a\left(x))u=f\left(x,u),& \hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,& \hspace{0.95em}| x| \to \infty ,\end{array}\right.where a0(x)+λa(x){a}_{0}\left(x)+\lambda a\left(x)is indefinite. By using a local linking theorem and the critical groups theory, they obtained the existence of solutions and their asymptotical behavior as λ→∞\lambda \to \infty .Xu and Chen [3] studied the following Kirchhoff problem: −a+b∫RN∣∇u∣2dxΔu+λV(x)u=f(x,u),x∈RN,u(x)∈H1(RN),\left\{\begin{array}{l}-\hspace{-0.25em}\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+\lambda V\left(x)u=f\left(x,u),\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right.where f(x,u)f\left(x,u)can be sublinear or superlinear. By using the genus theory, they obtained infinitely many negative solutions.Aleves et al. [4] dealt with the following Choquard equation: −Δu+(λa(x)+1)u=1∣x∣μ∗∣u∣p∣u∣p−2u,x∈RN,u(x)∈H1(RN),\left\{\begin{array}{l}-\hspace{-0.25em}\Delta u+\left(\lambda a\left(x)+1)u=\left(\frac{1}{| x\hspace{-0.25em}{| }^{\mu }}\ast | u\hspace{-0.25em}{| }^{p}\right)| u\hspace{-0.25em}{| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right.where μ∈(0,3)\mu \in \left(0,3), p∈(2,6−μ)p\in \left(2,6-\mu ), and the potential well Ω=⋃j=1kΩj\Omega ={\bigcup }_{j=1}^{k}{\Omega }_{j}. They proved the existence of a solution, which is nonzero on any subset Ωj{\Omega }_{j}. Furthermore, its asymptotical behavior was investigated.Zhao et al. [5] studied the Schrödinger-Poisson system allowing the potential V(x)V\left(x)changes sign −Δu+λV(x)u+K(x)ϕu=∣u∣p−2uinR3,−Δϕ=K(x)u2inR3,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\hspace{-0.25em}\Delta u+\lambda V\left(x)u+K\left(x)\phi u=| u\hspace{-0.25em}{| }^{p-2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.where p∈(3,6)p\in \left(3,6)and V∈C(R3,R)V\in C\left({{\mathbb{R}}}^{3},{\mathbb{R}})are bounded from below. Using the variational method, they obtained the existence and asymptotic behavior of nontrivial solutions.For the critical problems, Clapp and Ding [6] have studied the nonlinear Schrödinger equation: −Δu+λV(x)u=μu+u2∗−1,x∈RN\hspace{-18.75em}-\Delta u+\lambda V\left(x)u=\mu u+{u}^{{2}^{\ast }-1},\hspace{1.0em}x\in {{\mathbb{R}}}^{N}for N≥4,λ,μ>0N\ge 4,\lambda ,\mu \gt 0. By using variational methods, the authors established existence and multiplicity of positive solutions, which localize near the potential well for λ\lambda large and μ\mu small.Later, the corresponding results obtained in [6] were generalized to the fractional Schrödinger equations by Niu and Tang [7], where they have studied (−Δ)su+(λV(x)−μ)u=∣u∣2s∗−2u,x∈RN,u≥0,u∈Hs(RN).\hspace{0.45em}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\left(-\Delta )}^{s}u+\left(\lambda V\left(x)-\mu )u=| u\hspace{-0.25em}{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\ge 0,\hspace{1em}u\in {H}^{s}\left({{\mathbb{R}}}^{N}).\end{array}\right.Under the linear perturbation, [6] and [7] obtained the existence of solutions and their asymptotic behavior. For the nonlinear perturbation, Alves and Barros [8] considered −Δu+λV(x)u=μup−1+u2∗−1,x∈RN.-\Delta u+\lambda V\left(x)u=\mu {u}^{p-1}+{u}^{{2}^{\ast }-1},\hspace{1.0em}x\in {{\mathbb{R}}}^{N}.By employing the Ljusternik-Schnirelmann category, for λ\lambda big enough and μ\mu small enough, the aforementioned problem has at last cat(Ω){\rm{cat}}\left(\Omega )positive solutions.For more results about these kinds of problems and fractional Schrödinger equations, see, for example, [9,10, 11,12,13, 14,15,16, 17,18,19, 20,21,22, 23,24] and references therein. Motivated by the aforementioned results, we consider equation (1.1). The potential function V(x)V\left(x)satisfies (V1)V(x)∈C(RN,R)V\left(x)\in C\left({{\mathbb{R}}}^{N},{\mathbb{R}})such that V(x)≥0V\left(x)\ge 0, Ω≔intV−1(0)\Omega := {\rm{int}}{V}^{-1}\left(0)is a nonempty open set of class C0.1{C}^{0.1}with bounded boundary and V−1(0)=Ω¯{V}^{-1}\left(0)=\bar{\Omega };(V2)There exists M0>0{M}_{0}\gt 0such that μ({x∈RN:V(x)≤M0})<∞,\mu (\left\{x\in {{\mathbb{R}}}^{N}:V\left(x)\le {M}_{0}\right\})\lt \infty ,where μ\mu denotes the Lebesgue measure on RN{{\mathbb{R}}}^{N}.We first introdcue some notations. For s∈(0,1),p∈[1,+∞)s\in \left(0,1),p\in {[}1,+\infty ), define Ws,p(RN)≔u∈Lp(RN):∣u(x)−u(y)∣∣x−y∣Np+s∈Lp(RN×RN),\hspace{-39.85em}{W}^{s,p}\left({{\mathbb{R}}}^{N}):= \left\{\phantom{\rule[-1.25em]{}{0ex}},u\in {L}^{p}\left({{\mathbb{R}}}^{N}):\frac{| u\left(x)-u(y)| }{| x-y\hspace{-0.25em}{| }^{\tfrac{N}{p}+s}}\in {L}^{p}\left({{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N})\right\},endowed with the norm ‖u‖Ws,p(RN)≔∫RN∣u∣pdx+∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+spdxdy1p,\hspace{-40.35em}\Vert u{\Vert }_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}:= {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{p}{\rm{d}}x+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+sp}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}},where the term [u]Ws,p(RN)≔∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy1p\hspace{-40.1em}{\left[u]}_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}:= {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}is the so-called Gagliardo (semi)norm of uu. Moreover, we define Ws,p(Ω)≔u∈Lp(Ω):∣u(x)−u(y)∣∣x−y∣Np+s∈Lp(Ω×Ω),\hspace{-38.75em}{W}^{s,p}\left(\Omega ):= \left\{u\in {L}^{p}\left(\Omega ):\frac{| u\left(x)-u(y)| }{| x-y\hspace{-0.25em}{| }^{\tfrac{N}{p}+s}}\in {L}^{p}\left(\Omega \times \Omega )\right\},endowed with the norm ‖u‖Ws,p(Ω)≔∫Ω∣u∣pdx+∫Ω∫Ω∣u(x)−u(y)∣p∣x−y∣N+spdxdy1p.\hspace{-39.65em}\Vert u{\Vert }_{{W}^{s,p}\left(\Omega )}:= {\left(\mathop{\int }\limits_{\Omega }| u{| }^{p}{\rm{d}}x+\mathop{\int }\limits_{\Omega }\mathop{\int }\limits_{\Omega }\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+sp}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{1}{p}}.Let Eλ≔u∈Ws,p(RN):∫RNλV(x)∣u(x)∣pdx<∞,\hspace{-33.8em}{E}_{\lambda }:= \left\{u\in {W}^{s,p}\left({{\mathbb{R}}}^{N}):\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\lt \infty \right\},with the norm ‖u‖λ=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣u(x)∣pdx1p.\hspace{-35.5em}\Vert u{\Vert }_{\lambda }={\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{1}{p}}.The energy functional associated with (1.1) is (1.2)Jλ(u)=1p∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λp∫RNV(x)∣u(x)∣pdx−1q∫RNu+(x)qdxforu∈Eλ,{J}_{\lambda }\left(u)=\frac{1}{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{\lambda }{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\frac{1}{q}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}u\in {E}_{\lambda },where u+=max{u,0}{u}^{+}=\max \left\{u,0\right\}. Then, we can define the Nehari manifold ℳλ≔u∈Eλ⧹{0}:∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx=∫RNu+(x)qdx{{\mathcal{ {\mathcal M} }}}_{\lambda }:= \left\{u\in {E}_{\lambda }\setminus \left\{0\right\}:\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x=\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\right\}and cλ≔inf{Jλ(u):u∈ℳλ}.{c}_{\lambda }:= \inf \left\{{J}_{\lambda }\left(u):u\in {{\mathcal{ {\mathcal M} }}}_{\lambda }\right\}.Consider the following “limit” problem of (1.1) (1.3)(−Δ)psu(x)=u(x)q−1,x∈Ω,u(x)≥0,x∈Ω,u(x)=0,x∈RN⧹Ω,\left\{\begin{array}{l}{\left(-\Delta )}_{p}^{s}u\left(x)=u{\left(x)}^{q-1},\hspace{1em}x\in \Omega ,\\ u\left(x)\ge 0,\hspace{1em}x\in \Omega ,\\ u\left(x)=0,\hspace{1em}x\in {{\mathbb{R}}}^{N}\setminus \Omega ,\end{array}\right.Define a subspace E0{E}_{0}of Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})as follows: (1.4)E0≔{u∈Ws,p(RN):u(x)=0inRN⧹Ω}{E}_{0}:= \{u\in {W}^{s,p}\left({{\mathbb{R}}}^{N}):u\left(x)=0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega \}trΩE0={u∣Ω:u∈E0}.\hspace{-19.55em}{{\rm{tr}}}_{\Omega }{E}_{0}=\{u{| }_{\Omega }:u\in {E}_{0}\}.The energy functional associated with (1.3) can be defined by Φ(u)=1p∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy−1q∫Ωu+(x)qdxforu∈E0.\Phi \left(u)=\frac{1}{p}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y-\frac{1}{q}\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}u\in {E}_{0}.Then, the associated Nehari manifold is N≔u∈E0⧹{0}:∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy=∫Ωu+(x)qdx{\mathcal{N}}:= \left\{u\in {E}_{0}\setminus \left\{0\right\}:\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y=\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x\right\}and c(Ω)≔inf{Φ(u):u∈N}.c\left(\Omega ):= \inf \left\{\Phi \left(u):u\in {\mathcal{N}}\right\}.Definition 1.1A function uλ(x){u}_{\lambda }\left(x)is a ground state solution of (1.1) if cλ{c}_{\lambda }is achieved by uλ∈ℳλ{u}_{\lambda }\in {{\mathcal{ {\mathcal M} }}}_{\lambda }, which is a critical point of Jλ{J}_{\lambda }. Similarly, a function u(x)u\left(x)is a ground state solution of (1.3) if c(Ω)c\left(\Omega )is achieved by u∈Nu\in {\mathcal{N}}, which is a critical point of Φ\Phi .Definition 1.2Let XXbe a Banach space, φ∈C1(X,R).\varphi \in {C}^{1}\left(X,{\mathbb{R}}).The function φ\varphi satisfies the (PS)c{\left(PS)}_{c}condition if any sequence {un}⊆X\left\{{u}_{n}\right\}\subseteq X, such that (1.5)φ(un)→c,φ′(un)→0\varphi \left({u}_{n})\to c,\varphi ^{\prime} \left({u}_{n})\to 0has a convergent subsequence. The sequence {un}\left\{{u}_{n}\right\}that satisfies (1.5) is called to be a (PS)c{\left(PS)}_{c}sequence of φ\varphi .Our main results read as follows:Theorem 1.3Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold, then for λ\lambda large, (1.1) has a ground state solution uλ(x){u}_{\lambda }\left(x). Furthermore, any sequence λn→∞{\lambda }_{n}\to \infty , {uλn(x)}\left\{{u}_{{\lambda }_{n}}\left(x)\right\}has a subsequence such that uλn{u}_{{\lambda }_{n}}converges in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})along the subsequence to a ground state solution uuof (1.3).Theorem 1.4Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold. Let un,n∈N{u}_{n},n\in {\mathbb{N}}be a sequence of solutions of (1.1) with λ\lambda being replaced by λn{\lambda }_{n}(λn→∞{\lambda }_{n}\to \infty as n→∞n\to \infty ) such that lim supn→∞Jλ(un)<∞{\mathrm{lim\; sup}}_{n\to \infty }{J}_{\lambda }\left({u}_{n})\lt \infty . Then, un(x){u}_{n}\left(x)converges strongly along a subsequence in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})to a solution uuof (1.3).The following paper is organized as follows: In Section 2, we will give some preliminary results. Section 3 is devoted to the “limit” problem, and Section 4 contains the proofs of the main results. CCdenotes various generic positive constants, and o(1)o\left(1)will be used to represent quantities that tend to 0 as λ(orn)→∞\lambda \left(\hspace{0.1em}\text{or}\hspace{0.1em}\hspace{0.33em}n)\to \infty .2Preliminary resultsLemma 2.1Let λ0>0{\lambda }_{0}\gt 0be a fixed constant. Then, for λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, V(x)V\left(x)satisfying (V1)\left({V}_{1})and (V2)\left({V}_{2}), Eλ{E}_{\lambda }is continuously embedded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})uniformly in λ\lambda .ProofBy the definition of Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and Eλ{E}_{\lambda }, we only need to prove the following inequality: (2.1)∫RN∣u(x)∣pdx≤C∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣u(x)∣pdx.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u\left(x){| }^{p}{\rm{d}}x\le C\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| u\left(x){| }^{p}{\rm{d}}x\right).Define D≔{x∈RN:V(x)≤M0}\hspace{-12.4em}D:= \{x\in {{\mathbb{R}}}^{N}:V\left(x)\le {M}_{0}\}and Dδ0≔{x∈RN:dist(x,D)≤δ0}.{D}^{{\delta }_{0}}:= \{x\in {{\mathbb{R}}}^{N}:{\rm{dist}}\left(x,D)\le {\delta }_{0}\}.Take ζ∈C∞(RN,R)\zeta \in {C}^{\infty }\left({{\mathbb{R}}}^{N},{\mathbb{R}}), 0≤ζ≤10\le \zeta \le 1, satisfying (2.2)ζ(x)=1,x∈D,0,x∉Dδ0,∣∇ζ∣≤C/δ0.\hspace{-14.6em}\zeta \left(x)=\left\{\begin{array}{ll}1,& x\in D,\\ 0,& x\notin {D}^{{\delta }_{0}},\end{array}\right.\hspace{1.0em}| \nabla \zeta | \le C\hspace{-0.08em}\text{/}\hspace{-0.08em}{\delta }_{0}.Then, for any function u∈Eλu\in {E}_{\lambda }, we can obtain (2.3)∫RN(1−ζp)∣u(x)∣pdx=∫RN⧹D(1−ζp)∣u(x)∣pdx+∫D(1−ζp)∣u(x)∣pdx≤1λ0M0λ∫RNV(x)∣u(x)∣pdx\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x=\mathop{\int }\limits_{{{\mathbb{R}}}^{N}\setminus D}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x+\mathop{\int }\limits_{D}\left(1-{\zeta }^{p})| u\left(x){| }^{p}{\rm{d}}x\le \frac{1}{{\lambda }_{0}{M}_{0}}\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}xand (2.4)∫RNζp∣u(x)∣pdx=∫Dδ0ζp∣u(x)∣pdx≤μ(Dδ0)1−pps∗∫Dδ0∣u(x)∣ps∗dxpps∗≤Cμ(Dδ0)1−pps∗∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy,\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\zeta }^{p}| u\left(x){| }^{p}{\rm{d}}x& =& \mathop{\displaystyle \int }\limits_{{D}^{{\delta }_{0}}}{\zeta }^{p}| u\left(x){| }^{p}{\rm{d}}x\\ & \le & \mu {\left({D}^{{\delta }_{0}})}^{1-\tfrac{p}{{p}_{s}^{\ast }}}{\left(\mathop{\displaystyle \int }\limits_{{D}^{{\delta }_{0}}}| u\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{p}{{p}_{s}^{\ast }}}\\ & \le & C\mu {\left({D}^{{\delta }_{0}})}^{1-\tfrac{p}{{p}_{s}^{\ast }}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y,\end{array}where we have used (V2)\left({V}_{2})and the Sobolev trace inequality ∫RN∣u(x)∣ps∗dx1/ps∗≤C∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy1/p,{\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| u\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{1\text{/}{p}_{s}^{\ast }}\le C{\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{1\text{/}p},for u∈Ws,p(RN)u\in {W}^{s,p}\left({{\mathbb{R}}}^{N})and C=C(N,p,s)>0C=C\left(N,p,s)\gt 0. Thus, (2.1) follows from (2.3) and (2.4).□Lemma 2.2There exists σ>0\sigma \gt 0independent of λ\lambda , such that ‖u‖λ≥σ\Vert u{\Vert }_{\lambda }\ge \sigma for all u∈ℳλu\in {{\mathcal{ {\mathcal M} }}}_{\lambda }.ProofFrom Lemma 2.1, for any u∈ℳλu\in {{\mathcal{ {\mathcal M} }}}_{\lambda }, 0=⟨Jλ′(u),u⟩=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx−∫RNu+(x)qdx≥‖u‖λp−C‖u‖Ws,p(RN)q≥‖u‖λp−C‖u‖λq,\begin{array}{rcl}0=\langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle & =& \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\\ & \ge & \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{{W}^{s,p}\left({{\mathbb{R}}}^{N})}^{q}\\ & \ge & \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{\lambda }^{q},\end{array}where C>0C\gt 0is independent of λ≥0\lambda \ge 0. The aforementioned inequality implies that ‖u‖λq−p≥1C\Vert u{\Vert }_{\lambda }^{q-p}\ge \frac{1}{C}. Choosing σ=1C1q−p\sigma ={\left(\frac{1}{C}\right)}^{\tfrac{1}{q-p}}, we obtain ‖u‖λ≥σ\Vert u{\Vert }_{\lambda }\ge \sigma .□Lemma 2.3Let λ0{\lambda }_{0}be a fixed positive constant, there exists c0>0{c}_{0}\gt 0independent of λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, such that if {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }, then either c≥c0c\ge {c}_{0}or c=0c=0. Moreover, (2.5)lim supn→∞‖un‖λp≤pqq−pc.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \frac{pq}{q-p}c.ProofFrom the definition of (PS)c{\left(PS)}_{c}sequence, c+‖un‖λ⋅o(1)=Jλ(un)−1q⟨Jλ′(un),un⟩=1p−1q∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣un(x)∣pdx=q−ppq‖un‖λp.\begin{array}{rcl}c+\Vert {u}_{n}{\Vert }_{\lambda }\cdot o\left(1)& =& {J}_{\lambda }\left({u}_{n})-\frac{1}{q}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle \\ & =& \left(\frac{1}{p}-\frac{1}{q}\right)\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & =& \frac{q-p}{pq}\Vert {u}_{n}{\Vert }_{\lambda }^{p}.\end{array}Then, (2.5) holds. On the other side, there is a constant C>0C\gt 0independent of λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0, such that ⟨Jλ′(u),u⟩=∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy+λ∫RNV(x)∣u(x)∣pdx−∫RNu+(x)qdx≥‖u‖λp−C‖u‖λq.\langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle =\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\lambda \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| u\left(x){| }^{p}{\rm{d}}x-\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x\ge \Vert u{\Vert }_{\lambda }^{p}-C\Vert u{\Vert }_{\lambda }^{q}.Thus, there exists σ1>0{\sigma }_{1}\gt 0independent of λ\lambda , such that (2.6)14‖u‖λp≤⟨Jλ′(u),u⟩for‖u‖λ<σ1.\frac{1}{4}\Vert u{\Vert }_{\lambda }^{p}\le \langle {J}_{\lambda }^{^{\prime} }\left(u),u\rangle \hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}\Vert u{\Vert }_{\lambda }\lt {\sigma }_{1}.If c<σ1p(q−p)pqc\lt \frac{{\sigma }_{1}^{p}\left(q-p)}{pq}, then lim supn→∞‖un‖λp≤cpqq−p<σ1p.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \frac{cpq}{q-p}\lt {\sigma }_{1}^{p}.Hence, ‖un‖λ<σ1\Vert {u}_{n}{\Vert }_{\lambda }\lt {\sigma }_{1}for nnlarge. It follows from (2.6) that 14‖un‖λp≤⟨Jλ′(un),un⟩=o(1)‖un‖λ,\frac{1}{4}\Vert {u}_{n}{\Vert }_{\lambda }^{p}\le \langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle =o\left(1)\Vert {u}_{n}{\Vert }_{\lambda },which implies ‖un‖λ→0\Vert {u}_{n}{\Vert }_{\lambda }\to 0as n→∞n\to \infty . Therefore, Jλ(un)→0{J}_{\lambda }\left({u}_{n})\to 0, that is, c=0c=0. Thus, c0=σ1p(q−p)qp{c}_{0}=\frac{{\sigma }_{1}^{p}\left(q-p)}{qp}is as required.□Lemma 2.4There exists δ0>0{\delta }_{0}\gt 0, such that any (PS)c{\left(PS)}_{c}sequence {un}\left\{{u}_{n}\right\}of Jλ{J}_{\lambda }with λ≥0\lambda \ge 0and c>0c\gt 0satisfies(2.7)lim infn→∞‖un+‖Lq(RN)q≥δ0c.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\Vert {u}_{n}^{+}{\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q}\ge {\delta }_{0}c.ProofFrom the definition of (PS)c{\left(PS)}_{c}sequence, c=limn→∞Jλ(un)−1p⟨Jλ′(un),un⟩=1p−1qlimn→∞∫RNun+(x)qdx=(q−p)qplimn→∞‖un+(x)‖Lq(RN)q,c=\mathop{\mathrm{lim}}\limits_{n\to \infty }\left({J}_{\lambda }\left({u}_{n})-\frac{1}{p}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),{u}_{n}\rangle \right)=\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x=\frac{\left(q-p)}{qp}\mathop{\mathrm{lim}}\limits_{n\to \infty }\Vert {u}_{n}^{+}\left(x){\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q},which implies (2.7) with δ0≤qpq−p{\delta }_{0}\le \frac{qp}{q-p}.□Lemma 2.5Let C1{C}_{1}be any fixed constant. Then, for any ε>0\varepsilon \gt 0, there exists Λε>0{\Lambda }_{\varepsilon }\gt 0and Rε>0{R}_{\varepsilon }\gt 0, such that if {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }with λ≥Λε,c≤C1\lambda \ge {\Lambda }_{\varepsilon },c\le {C}_{1}, then(2.8)lim supn→∞∫BRεcun+(x)qdx≤ε,\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{{R}_{\varepsilon }}^{c}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x\le \varepsilon ,where BRεc={x∈RN:∣x∣≥Rε}{B}_{{R}_{\varepsilon }}^{c}=\left\{x\in {{\mathbb{R}}}^{N}:| x| \ge {R}_{\varepsilon }\right\}.ProofFor R>0R\gt 0, let A(R)≔{x∈RN:∣x∣>R,V(x)≥M0}A\left(R):= \left\{x\in {{\mathbb{R}}}^{N}:| x| \gt R,V\left(x)\ge {M}_{0}\right\}and B(R)≔{x∈RN:∣x∣>R,V(x)<M0}.\hspace{0.3em}B\left(R):= \left\{x\in {{\mathbb{R}}}^{N}:| x| \gt R,V\left(x)\lt {M}_{0}\right\}.It follows from Lemma 2.3 that (2.9)∫A(R)∣un(x)∣pdx≤1λM0∫RNλV(x)∣un(x)∣pdx≤1λM0∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+∫RNλV(x)∣un(x)∣pdx≤1λM0pqq−pC1+o(1).\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{A\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x& \le & \frac{1}{\lambda {M}_{0}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & \le & \frac{1}{\lambda {M}_{0}}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\lambda V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & \le & \frac{1}{\lambda {M}_{0}}\left(\frac{pq}{q-p}{C}_{1}+o\left(1)\right).\end{array}From Hölder inequality and (2.5), we can see that, for 1<r<N/(N−ps)1\lt r\lt N\hspace{-0.08em}\text{/}\hspace{-0.01em}\left(N-ps), (2.10)∫B(R)∣un(x)∣pdx≤∫RN∣un(x)∣prdx1/rμ(B(R))1/r′≤C‖un‖λp⋅μ(B(R))1/r′≤Cpqq−pC0⋅μ(B(R))1/r′,\mathop{\int }\limits_{B\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\le {\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {u}_{n}\left(x){| }^{pr}{\rm{d}}x\right)}^{1\text{/}r}\mu {(B\left(R))}^{1\text{/}r^{\prime} }\le C\Vert {u}_{n}{\Vert }_{\lambda }^{p}\cdot \mu {\left(B\left(R))}^{1\text{/}r^{\prime} }\le C\frac{pq}{q-p}{C}_{0}\cdot \mu {\left(B\left(R))}^{1\text{/}r^{\prime} },where C=C(N,r)>0C=C\left(N,r)\gt 0and 1/r+1/r′=11\hspace{0.1em}\text{/}r+1\text{/}\hspace{0.1em}r^{\prime} =1. By interpolation inequality and Sobolev embedding inequality, we can obtain ∫BRcun+(x)qdx≤∫BRc∣un(x)∣pdxq(1−θ)p⋅∫BRc∣un(x)∣ps∗dxqθps∗≤∫BRc∣un(x)∣pdxq(1−θ)p∫RN∣un(x)∣ps∗dxqθps∗≤C∫BRc∣un(x)∣pdxq(1−θ)p∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdyqθp≤C∫A(R)∣un(x)∣pdx+∫B(R)∣un(x)∣pdxq(1−θ)p‖un‖λqθ,\hspace{-28.55em}\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x& \le & {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}\cdot {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{q\theta }{{p}_{s}^{\ast }}}\\ & \le & {\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}{\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| {u}_{n}\left(x){| }^{{p}_{s}^{\ast }}{\rm{d}}x\right)}^{\tfrac{q\theta }{{p}_{s}^{\ast }}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{{B}_{R}^{c}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}{\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\right)}^{\tfrac{q\theta }{p}}\\ & \le & C{\left(\mathop{\displaystyle \int }\limits_{A\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x+\mathop{\displaystyle \int }\limits_{B\left(R)}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)}^{\tfrac{q\left(1-\theta )}{p}}\Vert {u}_{n}{\Vert }_{\lambda }^{q\theta },\end{array}where θ=Nsq−ppq\theta =\frac{N}{s}\frac{q-p}{pq}. Then, the result follows from (2.9), (2.10) and (V2)\left({V}_{2}).□Lemma 2.6(Brézis-Lieb lemma, 1983) Let {un}⊂Lp(RN)\left\{{u}_{n}\right\}\subset {L}^{p}\left({{\mathbb{R}}}^{N}), 1≤p<∞1\le p\lt \infty . If(a){un}\left\{{u}_{n}\right\}is bounded in Lp(RN){L}^{p}\left({{\mathbb{R}}}^{N}),(b)un→u{u}_{n}\to ualmost everywhere on RN{{\mathbb{R}}}^{N}, then (2.11)limn→∞(∣un∣pp−∣un−u∣pp)=∣u∣pp.\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(| {u}_{n}\hspace{-0.25em}{| }_{p}^{p}-| {u}_{n}-u\hspace{-0.25em}{| }_{p}^{p})=| u\hspace{-0.25em}{| }_{p}^{p}.Lemma 2.7Let λ≥λ0>0\lambda \ge {\lambda }_{0}\gt 0be fixed and let {un}\left\{{u}_{n}\right\}be a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }. Then, up to a subsequence, un⇀u{u}_{n}\rightharpoonup uin Eλ{E}_{\lambda }with uubeing a weak solution of (1.1). Moreover, un1=un−u{u}_{n}^{1}={u}_{n}-uis (PS)c′{\left(PS)}_{c^{\prime} }sequence with c′=c−Jλ(u)c^{\prime} =c-{J}_{\lambda }\left(u).ProofBy Lemma 2.3, {un}\left\{{u}_{n}\right\}is bounded in Eλ{E}_{\lambda }. Then, up to a subsequence un⇀u{u}_{n}\rightharpoonup uin Eλ{E}_{\lambda }as n→∞n\to \infty , and (2.12)un⇀uinWs,p(RN),\hspace{-15.95em}{u}_{n}\rightharpoonup u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{W}^{s,p}\left({{\mathbb{R}}}^{N}),(2.13)un⇀uinLq(RN),p≤q<ps∗,\hspace{-15.95em}{u}_{n}\rightharpoonup u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}^{q}\left({{\mathbb{R}}}^{N}),\hspace{1em}p\le q\lt {p}_{s}^{\ast },(2.14)un→uinLlocq(RN),p≤q<ps∗,{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}_{{\rm{loc}}}^{q}\left({{\mathbb{R}}}^{N}),\hspace{1em}p\le q\lt {p}_{s}^{\ast },(2.15)un→ua.e. inRN,{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{a.e. in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{N},where ps∗=NpN−ps{p}_{s}^{\ast }=\frac{Np}{N-ps}is the fractional critical Sobolev exponent. Hence, for any φ∈Eλ\varphi \in {E}_{\lambda }, we have ⟨Jλ′(un),φ⟩=∫RN∫RN∣un(x)−un(y)∣p−2(un(x)−un(y))(φ(x)−φ(y))∣x−y∣N+psdxdy+λ∫RNV(x)∣un(x)∣p−2un(x)φ(x)dx−∫RNun+(x)q−1φ(x)dx→∫RN∫RN∣u(x)−u(y)∣p−2(u(x)−u(y))(φ(x)−φ(y))∣x−y∣N+psdxdy+λ∫RNV(x)u(x)p−1φ(x)dx−∫RNu+(x)q−1φ(x)dx=⟨Jλ′(u),φ⟩.\begin{array}{rcl}\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle & =& \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y)\hspace{-0.25em}{| }^{p-2}\left({u}_{n}\left(x)-{u}_{n}(y))\left(\varphi \left(x)-\varphi (y))}{| x-y\hspace{-0.25em}\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p-2}{u}_{n}\left(x)\varphi \left(x){\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q-1}\varphi \left(x){\rm{d}}x\\ & \to & \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p-2}\left(u\left(x)-u(y))\left(\varphi \left(x)-\varphi (y))}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\lambda \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)u{\left(x)}^{p-1}\varphi \left(x){\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q-1}\varphi \left(x){\rm{d}}x=\langle {J}_{\lambda }^{^{\prime} }\left(u),\varphi \rangle .\end{array}Therefore, (2.16)⟨Jλ′(u),φ⟩=limn→∞⟨Jλ′(un),φ⟩=0,\langle {J}_{\lambda }^{^{\prime} }\left(u),\varphi \rangle =\mathop{\mathrm{lim}}\limits_{n\to \infty }\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle =0,which implies that uuis a critical point of Jλ{J}_{\lambda }.Let un1=un−u{u}_{n}^{1}={u}_{n}-u, we will show that as n→∞n\to \infty , (2.17)Jλ(un1)→c−Jλ(u){J}_{\lambda }\left({u}_{n}^{1})\to c-{J}_{\lambda }\left(u)and (2.18)Jλ′(un1)→0.{J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1})\to 0.To show (2.17), we observe that (2.19)Jλ(un1)=1p∫RN∫RN∣un1(x)−un1(y)∣p∣x−y∣N+psdxdy+λp∫RNV(x)∣un1(x)∣pdx−1q∫RNun1+(x)qdx=Jλ(un)−Jλ(u)+λp∫RNV(x)(∣un1(x)∣p−∣un(x)∣p+∣u(x)∣p)dx+1p∫RN∫RN∣un1(x)−un1(y)∣p−∣un(x)−un(y)∣p+∣u(x)−u(y)∣p∣x−y∣N+psdxdy+1q∫RNun+(x)qdx−1q∫RNun1+(x)qdx−1q∫RNu+(x)qdx.\begin{array}{rcl}{J}_{\lambda }\left({u}_{n}^{1})& =& \frac{1}{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}^{1}\left(x)-{u}_{n}^{1}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{\lambda }{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}^{1}\left(x){| }^{p}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x\\ & =& {J}_{\lambda }\left({u}_{n})-{J}_{\lambda }\left(u)+\frac{\lambda }{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)\left(| {u}_{n}^{1}\left(x){| }^{p}-| {u}_{n}\left(x){| }^{p}+| u\left(x){| }^{p}){\rm{d}}x\\ & & +\frac{1}{p}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}^{1}\left(x)-{u}_{n}^{1}(y){| }^{p}-| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}+| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & & +\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x-\frac{1}{q}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x.\end{array}From Lemma 2.6, ∫RNun+(x)qdx−∫RNu+(x)qdx−∫RNun1+(x)qdx→0{\int }_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-{\int }_{{{\mathbb{R}}}^{N}}{u}^{+}{\left(x)}^{q}{\rm{d}}x-{\int }_{{{\mathbb{R}}}^{N}}{u}_{n}^{1+}{\left(x)}^{q}{\rm{d}}x\to 0as n→∞n\to \infty . Conversely, we know that ‖un‖λp−‖u‖λp−‖un1‖λp→0\Vert {u}_{n}{\Vert }_{\lambda }^{p}-\Vert u{\Vert }_{\lambda }^{p}-\Vert {u}_{n}^{1}{\Vert }_{\lambda }^{p}\to 0, as n→∞n\to \infty . Thus, from (2.19), we indeed have obtained (2.17). Now we come to show (2.18). From (2.16), we have for any φ∈Eλ\varphi \in {E}_{\lambda }⟨Jλ′(un1),φ⟩=⟨Jλ′(un),φ⟩−∫RN(un1+)q−1φ(x)dx+∫RN(un+)q−1φ(x)dx−∫RN(u+)q−1φ(x)dx+o(1).\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1}),\varphi \rangle =\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}),\varphi \rangle -\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}_{n}^{1+})}^{q-1}\varphi \left(x){\rm{d}}x+\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}_{n}^{+})}^{q-1}\varphi \left(x){\rm{d}}x-\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{\left({u}^{+})}^{q-1}\varphi \left(x){\rm{d}}x+o\left(1).Since Jλ′(un)→0{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to 0and un⇀u{u}_{n}\rightharpoonup uin Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N}), we have limn→∞sup‖φ‖λ≤1∫RN((un1+)q−1(x)φ(x)−(un+)q−1φ(x)+(u+)q−1φ(x))dx=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\sup }\limits_{\Vert \varphi {\Vert }_{\lambda }\le 1}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}({\left({u}_{n}^{1+})}^{q-1}\left(x)\varphi \left(x)-{\left({u}_{n}^{+})}^{q-1}\varphi \left(x)+{\left({u}^{+})}^{q-1}\varphi \left(x)){\rm{d}}x=0.Thus, we have limn→∞⟨Jλ′(un1),φ⟩=0for anyφ∈Eλ,\mathop{\mathrm{lim}}\limits_{n\to \infty }\langle {J}_{\lambda }^{^{\prime} }\left({u}_{n}^{1}),\varphi \rangle =0\hspace{0.33em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}\varphi \in {E}_{\lambda },which implies (2.18), and this completes the proof.□Proposition 2.8Suppose (V1)\left({V}_{1})and (V2)\left({V}_{2})hold. Then, for any C0>0{C}_{0}\gt 0, there exists Λ0>0{\Lambda }_{0}\gt 0such that Jλ{J}_{\lambda }satisfies the (PS)c{\left(PS)}_{c}condition for all λ≥Λ0\lambda \ge {\Lambda }_{0}and c≤C0c\le {C}_{0}.ProofChoose 0<ε<δ0c0/20\lt \varepsilon \lt {\delta }_{0}{c}_{0}\hspace{-0.08em}\text{/}\hspace{-0.08em}2, where c0{c}_{0}and δ0{\delta }_{0}are the constants in Lemmas 2.3 and 2.4, respectively. Let Λ0≔Λε{\Lambda }_{0}:= {\Lambda }_{\varepsilon }, where Λε>0{\Lambda }_{\varepsilon }\gt 0is from Lemma 2.5.Assume {un}\left\{{u}_{n}\right\}is a (PS)c{\left(PS)}_{c}sequence of Jλ{J}_{\lambda }with λ≥Λ0\lambda \ge {\Lambda }_{0}and c≤C0c\le {C}_{0}. By Lemma 2.7, un1=un−u{u}_{n}^{1}={u}_{n}-uis a (PS)c′{\left(PS)}_{c^{\prime} }sequence of Jλ{J}_{\lambda }with c′=c−Jλ(u)c^{\prime} =c-{J}_{\lambda }\left(u). If c′>0c^{\prime} \gt 0, it follows from Lemma 2.3 that c′≥c0c^{\prime} \ge {c}_{0}. From Lemma 2.4, we can obtain lim infn→∞‖un1+(⋅)‖Lq(RN)q≥δ0c′≥δ0c0.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\Vert {u}_{n}^{1+}\left(\cdot ){\Vert }_{{L}^{q}\left({{\mathbb{R}}}^{N})}^{q}\ge {\delta }_{0}c^{\prime} \ge {\delta }_{0}{c}_{0}.Conversely, Lemma 2.5 implies lim supn→∞∫BRεcun1+(x)q≤ε<δ0c02.\mathop{\mathrm{lim\; sup}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{{R}_{\varepsilon }}^{c}}{u}_{n}^{1+}{\left(x)}^{q}\le \varepsilon \lt \frac{{\delta }_{0}{c}_{0}}{2}.Noting un1→0{u}_{n}^{1}\to 0in Llocq(RN){L}_{{\rm{loc}}}^{q}\left({{\mathbb{R}}}^{N}), p≤q<ps∗p\le q\lt {p}_{s}^{\ast }, a contradiction follows from the aforementioned two inequalities. Therefore, c′=0c^{\prime} =0. Thus, un1→0{u}_{n}^{1}\to 0in Eλ{E}_{\lambda }by Lemma 2.3.□Corollary 2.9For any q∈(p,ps∗)q\in \left(p,{p}_{s}^{\ast }), there exists Λ0>0{\Lambda }_{0}\gt 0, such that cλ{c}_{\lambda }is achieved for all λ≥Λ0\lambda \ge {\Lambda }_{0}at some uλ∈Eλ{u}_{\lambda }\in {E}_{\lambda }, which is a ground state solution of (1.1).ProofBy Ekeland variational principle, there is a PS sequence un∈Eλ{u}_{n}\in {E}_{\lambda }, such that Jλ(un)→cλandJλ′(un)→0.{J}_{\lambda }\left({u}_{n})\to {c}_{\lambda }\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to 0.By Proposition 2.8, there exists some uλ∈Eλ{u}_{\lambda }\in {E}_{\lambda }, such that, up to subsequence, un→uλ{u}_{n}\to {u}_{\lambda }in Eλ{E}_{\lambda }as n→∞n\to \infty and λ\lambda is sufficiently large. It is not difficult to show that Jλ(un)→Jλ(uλ)andJλ′(un)→Jλ′(uλ).{J}_{\lambda }\left({u}_{n})\to {J}_{\lambda }\left({u}_{\lambda })\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{J}_{\lambda }^{^{\prime} }\left({u}_{n})\to {J}_{\lambda }^{^{\prime} }\left({u}_{\lambda }).Therefore, we have Jλ(uλ)=cλandJλ′(uλ)=0.{J}_{\lambda }\left({u}_{\lambda })={c}_{\lambda }\hspace{0.25em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.25em}{J}_{\lambda }^{^{\prime} }\left({u}_{\lambda })=0.This means that uλ{u}_{\lambda }is a ground state solution of (1.1).□3Limit problemLemma 3.1Let 1<p<q<ps∗≔pNN−ps1\lt p\lt q\lt {p}_{s}^{\ast }:= \frac{pN}{N-ps}, N≥2N\ge 2. Then, trΩE0t{r}_{\Omega }{E}_{0}is compactly embedded in Lq(Ω){L}^{q}\left(\Omega ).ProofSince trΩE0⊂Ws,p(Ω)t{r}_{\Omega }{E}_{0}\subset {W}^{s,p}\left(\Omega )and Ws,p(Ω)↪Lp(Ω){W}^{s,p}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{p}\left(\Omega )are compact for p<q<ps∗p\lt q\lt {p}_{s}^{\ast }, N≥2N\ge 2, the result follows.□Lemma 3.2The infimum c(Ω)c\left(\Omega )is achieved by a function u∈Nu\in {\mathcal{N}}, which is a ground state solution of (1.3).ProofBy Ekeland variational principle, there is a PS sequence un∈E0{u}_{n}\in {E}_{0}, such that Φ(un)→c(Ω)andΦ′(un)→0.\Phi \left({u}_{n})\to c\left(\Omega )\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\Phi ^{\prime} \left({u}_{n})\to 0.Thus, by Lemma 3.1, we can easily obtain a subsequence of {un}\left\{{u}_{n}\right\}(still denote it itself), such that un→u{u}_{n}\to uin E0{E}_{0}. Therefore, uuis a ground state solution of (1.3).□Remark 3.3Assume set Ω=intV−1(0)\Omega ={\rm{int}}{V}^{-1}\left(0)has more than one isolated component, for example, Ω=Ω1∪Ω2\Omega ={\Omega }_{1}\cup {\Omega }_{2}with Ω1∩Ω2=∅{\Omega }_{1}\cap {\Omega }_{2}=\varnothing . Suppose that u∈Nu\in {\mathcal{N}}is a nonnegative solution of (1.3) with u(x)=0u\left(x)=0in Ω1{\Omega }_{1}and u(x)≩0u\left(x)\gneqq 0in Ω2{\Omega }_{2}. Then, we have (−Δ)psu(x)=∫RN∣u(x)−u(y)∣p−2(u(x)−u(y))∣x−y∣N+psdy<0{\left(-\Delta )}_{p}^{s}u\left(x)={\int }_{{{\mathbb{R}}}^{N}}\frac{| \hspace{-0.25em}u\left(x)-u(y)\hspace{-0.25em}{| }^{p-2}\left(u\left(x)-u(y))}{| \hspace{-0.25em}x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}y\lt 0in Ω1{\Omega }_{1}. Conversely, (−Δ)psu(x)=u(x)q−1=0{\left(-\Delta )}_{p}^{s}u\left(x)=u{\left(x)}^{q-1}=0for x∈Ω1x\in {\Omega }_{1}. This contradiction shows that the nonnegative solution u(x)u\left(x)of (1.3) must be u(x)≩0u\left(x)\gneqq 0in both Ω1{\Omega }_{1}and Ω2.{\Omega }_{2}.However, the Laplacian case can have a nonnegative solution u(x)u\left(x)satisfying u(x)=0u\left(x)=0in Ω1{\Omega }_{1}and u(x)≩0u\left(x)\gneqq 0in Ω2{\Omega }_{2}. The difference between the two phenomena is attributed to the nonlocality of fractional operators and the locality of Laplacian operators.4The proof of the main resultsLemma 4.1cλ→c(Ω){c}_{\lambda }\to c\left(\Omega )as λ→∞\lambda \to \infty .ProofFrom the definition of cλ{c}_{\lambda }and c(Ω)c\left(\Omega ), we know that cλ≤c(Ω){c}_{\lambda }\le c\left(\Omega ), λ>0\lambda \gt 0. Furthermore, cλ{c}_{\lambda }is monotone increasing about the parameter λ>0.\lambda \gt 0.Then, there exists a constant kk, such that limn→∞cλn=k,\mathop{\mathrm{lim}}\limits_{n\to \infty }{c}_{{\lambda }_{n}}=k,where λn→∞{\lambda }_{n}\to \infty . It follows from Lemma 2.3 that k>0k\gt 0. By Corollary 2.9, for nnlarge enough, there exists a sequence un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}, such that Jλn′(un)=0{J}_{{\lambda }_{n}}^{^{\prime} }\left({u}_{n})=0and Jλn(un)=cλn.{J}_{{\lambda }_{n}}\left({u}_{n})={c}_{{\lambda }_{n}}.If k<c(Ω)k\lt c\left(\Omega ), it is easy to see that {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}); thus, we can assume that un⇀u{u}_{n}\rightharpoonup uin EEand (4.1)un(x)→u(x)inLlocθ(RN)forp≤θ<ps∗.{u}_{n}\left(x)\to u\left(x)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})\hspace{1em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}p\le \theta \lt {p}_{s}^{\ast }.Claim 1: u∣Ωc=0u{| }_{{\Omega }^{c}}=0. In fact, if u∣Ωc≠0u\hspace{-0.25em}{| }_{{\Omega }^{c}}\ne 0, then there exists a compact subset F⊂ΩcF\subset {\Omega }^{c}with dist(F,Ω)>0\left(F,\Omega )\gt 0, such that u∣F≠0u{| }_{F}\ne 0. It follows from (4.1) that ∫F∣un(x)∣pdx→∫F∣u(x)∣pdx>0.\mathop{\int }\limits_{F}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to \mathop{\int }\limits_{F}| u\left(x){| }^{p}{\rm{d}}x\gt 0.However, there exists ε0>0{\varepsilon }_{0}\gt 0, such that V(x)≥ε0>0V\left(x)\ge {\varepsilon }_{0}\gt 0, x∈Fx\in F. Thus, Jλn(un)≥q−ppqλn∫FV(x)∣un(x)∣pdx≥q−ppqλnε0∫F∣un(x)∣pdx→∞asn→∞,{J}_{{\lambda }_{n}}\left({u}_{n})\ge \frac{q-p}{pq}{\lambda }_{n}\mathop{\int }\limits_{F}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\ge \frac{q-p}{pq}{\lambda }_{n}{\varepsilon }_{0}\mathop{\int }\limits_{F}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to \infty \hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,which is a contradiction. Therefore, u∈E0u\in {E}_{0}.Claim 2: un→u{u}_{n}\to uin Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N})for p<q<ps∗p\lt q\lt {p}_{s}^{\ast }. Indeed, if not, then by the concentration-compactness lemma from the study by Loins [25], there exist δ>0,ρ>0\delta \gt 0,\rho \gt 0and xn∈RN{x}_{n}\in {{\mathbb{R}}}^{N}with ∣xn∣→∞| {x}_{n}| \to \infty , such that lim infn→∞∫Bρ(xn)∣un(x)−u(x)∣pdx≥δ>0.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\mathop{\int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\ge \delta \gt 0.Then, we have Jλn(un)=q−ppq∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+q−ppq∫RNλnV(x)∣un(x)∣pdx≥q−ppqλn∫Bρ(xn)∩{x:V(x)≥M0}V(x)∣un(x)∣pdx=q−ppqλn∫Bρ(xn)∩{x:V(x)≥M0}V(x)∣un(x)−u(x)∣pdx≥q−ppqλnM0∫Bρ(xn)∣un(x)−u(x)∣pdx−M0∫Bρ(xn)∩{x:V(x)≤M0}∣un(x)∣pdx≥q−ppqλnM0∫Bρ(xn)∣un(x)−u(x)∣pdx−o(1)→∞asn→∞,\begin{array}{rcl}{J}_{{\lambda }_{n}}\left({u}_{n})& =& \frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\ge {M}_{0}\right\}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\\ & =& \frac{q-p}{pq}{\lambda }_{n}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\ge {M}_{0}\right\}}V\left(x)| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\left({M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x-{M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})\cap \left\{x:V\left(x)\le {M}_{0}\right\}}| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & \ge & \frac{q-p}{pq}{\lambda }_{n}\left({M}_{0}\mathop{\displaystyle \int }\limits_{{B}_{\rho }\left({x}_{n})}| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x-o\left(1)\right)\to \infty \hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,\end{array}as a contradiction. So un→u{u}_{n}\to uin Lp(RN){L}^{p}\left({{\mathbb{R}}}^{N}). Therefore, it is easy to see that u≥0u\ge 0is a solution for problem (1.3). Furthermore, k=limn→∞cλn=limn→∞Jλn(un)=limn→∞1p−1q∫RNun+(x)qdx=1p−1q∫Ωu+(x)qdx,k=\mathop{\mathrm{lim}}\limits_{n\to \infty }{c}_{{\lambda }_{n}}=\mathop{\mathrm{lim}}\limits_{n\to \infty }{J}_{{\lambda }_{n}}\left({u}_{n})=\mathop{\mathrm{lim}}\limits_{n\to \infty }\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x=\left(\frac{1}{p}-\frac{1}{q}\right)\mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x,which means u∈Nu\in {\mathcal{N}}, and then, k≥c(Ω)k\ge c\left(\Omega ), a contradiction. Hence, limλ→∞cλ=c(Ω){\mathrm{lim}}_{\lambda \to \infty }\hspace{0.25em}{c}_{\lambda }=c\left(\Omega ).□Proof of Theorem 1.3By Corollary 2.9, there exists un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}, such that Jλn(un)=cλn{J}_{{\lambda }_{n}}\left({u}_{n})={c}_{{\lambda }_{n}}(λn→∞asn→∞{\lambda }_{n}\to \infty \hspace{0.25em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.25em}n\to \infty ). It is easy to see that {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). Then, without loss of generality, un⇀u{u}_{n}\rightharpoonup uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and un→u{u}_{n}\to uin Llocθ(RN){L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗p\lt \theta \lt {p}_{s}^{\ast }.Now we prove that un→u{u}_{n}\to ustrongly in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and uuis a ground state solution of (1.3). First, as the proof of Lemma 4.1, u≥0u\ge 0is a solution of problem (1.3) and un+→u+{u}_{n}^{+}\to {u}^{+}strongly in Lq(RN){L}^{q}\left({{\mathbb{R}}}^{N}).Now we claim that λn∫RNV(x)∣un(x)∣pdx→0{\lambda }_{n}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\to 0and ∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy→∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\to \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y.Indeed, if either lim infn→∞λn∫RNV(x)∣un(x)∣pdx>0\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }{\lambda }_{n}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\gt 0or lim infn→∞∫RN∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy>∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy.\mathop{\mathrm{lim\; inf}}\limits_{n\to \infty }\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\gt \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y.Thus, we have ∫RN∫RN∣u(x)−u(y)∣p∣x−y∣N+psdxdy<∫Ωu+(x)qdx.\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\lt \mathop{\int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x.Therefore, there is α∈(0,1)\alpha \in \left(0,1), such that αu∈N\alpha u\in {\mathcal{N}}and c(Ω)≤Φ(αu)=q−ppq∫RN∫RN∣αu(x)−αu(y)∣p∣x−y∣N+psdxdy<q−ppq∫RN∫RN∣u(x)−u(y)∣p∣x−y∣n+psdxdy≤limn→∞q−ppq∫RN∣un(x)−un(y)∣p∣x−y∣N+psdxdy+∫RNλnV(x)∣un(x)∣pdx=limn→∞Jλn(un)=c(Ω),\begin{array}{rcl}c\left(\Omega )& \le & \Phi \left(\alpha u)=\frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| \alpha u\left(x)-\alpha u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y\\ & \lt & \frac{q-p}{pq}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y\\ & \le & \mathop{\mathrm{lim}}\limits_{n\to \infty }\frac{q-p}{pq}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{N+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x\right)\\ & =& \mathop{\mathrm{lim}}\limits_{n\to \infty }{J}_{{\lambda }_{n}}\left({u}_{n})=c\left(\Omega ),\end{array}which is a contradiction. By now we complete the proof of Theorem 1.3.□Proof of Theorem 1.4Suppose {un}⊂Ws,p(RN)\left\{{u}_{n}\right\}\subset {W}^{s,p}\left({{\mathbb{R}}}^{N})is a solution of (1.1) with λ\lambda being replaced by λn{\lambda }_{n}(λn→∞{\lambda }_{n}\to \infty as n→∞n\to \infty ). It follows from lim supn→∞Jλ(un)<∞{\mathrm{lim\; sup}}_{n\to \infty }{J}_{\lambda }\left({u}_{n})\lt \infty that such a sequence {un}\left\{{u}_{n}\right\}is bounded in Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). Suppose that un⇀u{u}_{n}\rightharpoonup uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N})and un→u{u}_{n}\to uin Llocθ(RN){L}_{{\rm{loc}}}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗.p\lt \theta \lt {p}_{s}^{\ast }.Similar to the proof of Lemma 4.1, u∣Ωc=0u{| }_{{\Omega }^{c}}=0and u∈E0u\in {E}_{0}is solution of (1.3). Moreover, un→u{u}_{n}\to uin Lθ(RN){L}^{\theta }\left({{\mathbb{R}}}^{N})for p<θ<ps∗p\lt \theta \lt {p}_{s}^{\ast }. Noting un∈ℳλn{u}_{n}\in {{\mathcal{ {\mathcal M} }}}_{{\lambda }_{n}}and u∈Nu\in {\mathcal{N}}, we can obtain ∫RN∫RN∣un(x)−un(y)−u(x)+u(y)∣p∣x−y∣n+psdxdy+∫RNλnV(x)∣un(x)−u(x)∣pdx=∫RN∫RN∣un(x)−un(y)∣p∣x−y∣n+psdxdy+∫RNλnV(x)∣un(x)∣pdx−∫RN∫RN∣u(x)−u(y)∣p∣x−y∣n+psdxdy−∫RNλnV(x)∣u(x)∣pdx+o(1)=∫RNun+(x)qdx−∫Ωu+(x)qdx+o(1)=o(1).\begin{array}{l}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y)-u\left(x)+u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x)-u\left(x){| }^{p}{\rm{d}}x\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| {u}_{n}\left(x)-{u}_{n}(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y+\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| {u}_{n}\left(x){| }^{p}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u\left(x)-u(y){| }^{p}}{| x-y\hspace{-0.25em}{| }^{n+ps}}{\rm{d}}x{\rm{d}}y-\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\lambda }_{n}V\left(x)| u\left(x){| }^{p}{\rm{d}}x+o\left(1)\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}_{n}^{+}{\left(x)}^{q}{\rm{d}}x-\mathop{\displaystyle \int }\limits_{\Omega }{u}^{+}{\left(x)}^{q}{\rm{d}}x+o\left(1)=o\left(1).\end{array}Thus, un→u{u}_{n}\to uin Ws,p(RN){W}^{s,p}\left({{\mathbb{R}}}^{N}). This completes the proof of Theorem 1.4.□

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: nonlinear Schrödinger equation; ground state solution; fractional p-Laplacian; variational methods; 35J60; 35B33

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