Multiplicity of Positive Solutions for Schrödinger–Poisson Systems with a Critical Nonlinearity in $$\mathbb {R}^3$$ R 3

Abstract

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrodinger–Poisson problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\phi u=\lambda Q(x)|u|^{q-2}u+ K(x)|u|^4u, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ -\Delta \phi =u^2, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ \end{array}\right. \end{aligned}$$ where $$4<q<6 $$ or $$q=2$$ , $$\lambda >0$$ is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small $$\lambda $$ .