This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ - ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + μ ϕ | u | p - 2 u = f ( x , u ) + g ( x , u ) , in R 3 , ( - Δ ) α 2 ϕ = μ | u | p , in R 3 , where $$a>0,~b,~\mu \ge 0$$ a > 0 , b , μ ≥ 0 are constants, $$\alpha \in (0,3)$$ α ∈ ( 0 , 3 ) , $$p\in [2,3+2\alpha )$$ p ∈ [ 2 , 3 + 2 α ) , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ ϕ | u | p - 2 u is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.
Mediterranean Journal of Mathematics – Springer Journals
Published: May 28, 2018
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