Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

Abstract

This paper is concerned with the following Kirchhoff-type equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$ where $$a>0,~b,~\mu \ge 0$$ are constants, $$\alpha \in (0,3)$$ , $$p\in [2,3+2\alpha )$$ , the potential V(x) may be unbounded from below and $$\phi |u|^{p-2}u$$ 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 Schrodinger equations, the Kirchhoff equations and the Schrodinger–Poisson system.