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

We study the following critical Schrödinger-Possion system with steep potential well \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u+(1+\lambda V(x))u+\phi u=f(u)+|u|^4u,&\text {in}\ {\mathbb {R}}^{3},\\&-\Delta \phi =u^2,&\text {in}\ {\mathbb {R}}^{3}, \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& -\mathrm{\Delta }u+(1+\lambda V(x))u+\varphi u=f(u)+{|u|}^{4}u,\hfill & \hfill \text{in}{\mathbb{R}}^3,\\ \hfill & -\mathrm{\Delta }\varphi =u^2,\hfill & \hfill \text{in}{\mathbb{R}}^3,\end{array}\right)\end{array}$$ where \(\lambda >0\) $\lambda >0$ is a positive parameter, \(V:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) $V:{\mathbb{R}}^3\to \mathbb{R}$ is a continuous function and f is a continuous subcritical nonlinearity. Under some certain assumptions on V and f, for any \(\lambda \ge \lambda _0>0\) $\lambda \ge {\lambda }_0>0$ , 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 \(\lambda \rightarrow \infty \) $\lambda \to \infty$ . Our results extends that in Jiang[11](J. Differ. Equ. 2011) and Zhao[21](J. Differ. Equ. 2013) to the critical growth case.