Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential

Abstract

In this paper, we study the existence and multiplicity of solutions for the Schrodinger–Poisson equations $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda V(x)u+K(x)\phi u=f(x,u)\ \ \ \ \ &{} \ \text{ in }\mathbb {R}^3,\\ -\Delta \phi =K(x)u^2\ \ \ \ \ \ &{} \ \text{ in } \mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(\lambda >0\) is a parameter, the potential \(V\) may change sign and \(f\) is either superlinear or sublinear in \(u\) as \(|u|\rightarrow \infty \).