Multiplicity of solutions for a class of critical Schr\"odinger-Poisson system with two parameters

Abstract

We study a class of critical Schr\"odinger-Poisson system of the form \begin{equation*} \begin{cases} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -\Delta \phi=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases} \end{equation*} where $\lambda, \mu>0$ are two parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of positive ground state solutions for $\lambda$ large enough and $\mu>0$, and their asymptotical behavior as $\lambda\to\infty$. Moreover, by using Ljusternik-Schnirelmann theory, we obtain the existence of multiple positive solutions if $\lambda$ is large and $\mu$ is small.