Multiple positive solutions for a Schrödinger-Poisson system with critical and supercritical growths

Abstract

In this paper, we are concerned with the following Schrödinger-Poisson system $$ \begin{cases} -\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu \dfrac{|x|^\beta}{1+|x|^\beta}|u|^{q-2}u&in \mathbb{R}^3, -\Delta \phi=u^{2} &in \mathbb{R}^3, \end{cases} $$ where $0< \beta<3$, $6<q<6+2\beta$, $Q(x)$ is a positive continuous function on $\mathbb{R}^3$, $\lambda,\mu>0$ are real parameters. By the variational method and the Nehari method, we obtain that the system has $k$ positive solutions.