Existence and multiplicity of positive solutions for a Schrödinger-Poisson system with a perturbation

Abstract

In this paper we study the nonlinear Schrödinger-Poisson system with a perturbation: \begin{equation*} \begin{cases} -\Delta u+u+K( x) \phi u=\vert u\vert ^{p-2}u+\lambda f(x)\vert u\vert ^{q-2}u & \text{in }\mathbb{R}^{3}, \\ -\Delta \phi =K( x) u^{2} & \text{in }\mathbb{R}^{3}, \end{cases} \end{equation*} where $K$ and $f$ are nonnegative functions, $2\le q\leq p\le 6$ and $p\ge 4$, and the parameter $\lambda \in \mathbb{R}$. Under some suitable assumptions on $K $ and $f$, the criteria of existence and multiplicity of positive solutions are established by means of the Lusternik-Schnirelmann category and minimax method.