Existence and concentration of solutions for nonautomous Schrödinger–Poisson systems with critical growth

Abstract

In this paper, we study the following Schrödinger–Poisson system \begin{equation*} \begin{cases} -\Delta u+u+\mu \phi u=\lambda f(x,u)+u^5\quad & \mbox{in }\mathbb{R}^3,\\ -\Delta \phi=\mu u^2\quad & \mbox{in }\mathbb{R}^3, \end{cases} \end{equation*} where $\mu$, $\lambda>0$ are parameters and $f\in C(\mathbb{R}^3\times \mathbb{R},\mathbb{R})$. Under certain general assumptions on $f(x,u)$, we prove the existence and concentration of solutions of the above system for each $\mu>0$ and $\lambda$ sufficiently large. Our main result can be viewed as an extension of the results by Zhang [Nonlinear Anal. 75(2012), 6391–6401].

Keywords

• schrödinger–poisson system
• critical growth
• variational methods