Ground state solutions to critical Schrödinger–Possion system with steep potential well

Abstract

We study the following critical Schrödinger-Possion system with steep potential well

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+(1+\lambda V(x))u+\phi u=f(u)+|u|^4u,&\text {in}\ {\mathbb {R}}^{3},\\&-\Delta \phi =u^2,&\text {in}\ {\mathbb {R}}^{3}, \end{aligned}\right. \end{aligned}$$

where \(\lambda >0\) is a positive parameter, \(V:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) is a continuous function and f is a continuous subcritical nonlinearity. Under some certain assumptions on V and f, for any \(\lambda \ge \lambda _0>0\), we prove the existence of a ground state solution via variational methods. Moreover, the concentration behavior of the ground state solution is also described as \(\lambda \rightarrow \infty \). Our results extends that in Jiang[11](J. Differ. Equ. 2011) and Zhao[21](J. Differ. Equ. 2013) to the critical growth case.