Existence and Concentration of Ground State Solutions for a Schrödinger–Poisson-Type System with Steep Potential Well

Abstract

In this paper, we study the following nonlocal problem in \(\mathbb R^3\)

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

where \(\lambda >0\) is a real parameter and \(\mu >0\) is small enough. Under some suitable assumptions on V(x) and f(x, u), we prove the existence of ground state solutions for the problem when \(\lambda \) is large enough via variational methods. In addition, the concentration behavior of these ground state solutions is also investigated as \(\lambda \rightarrow +\infty \).