Multiplicity of solutions for a class of critical Schrödinger-Poisson system with two parameters

Abstract

We study a class of critical Schrödinger-Poisson system of the form \begin{equation*} \begin{cases} -Δu+λV(x)u+ϕu=μ|u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -Δϕ=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases} \end{equation*} where $λ, μ>0$ 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 $λ$ large enough and $μ>0$, and their asymptotical behavior as $λ\to\infty$. Moreover, by using Ljusternik-Schnirelmann theory, we obtain the existence of multiple positive solutions if $λ$ is large and $μ$ is small.