Existence and concentration behavior of solutions for the logarithmic Schrödinger–Poisson system with steep potential

Abstract

In this paper, we study the following logarithmic Schrödinger–Poisson system:

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

where $\lambda >0$, $V(x)\in C({\mathbb {R}}^3,{\mathbb {R}})$ and $q(x)\ge 0$ for all $x\in {\mathbb {R}}^3$. Under the suitable conditions on potential V(x) and q(x), by proceeding a new penalization scheme to the nonlocal term $\phi$, combining with a new version of Minimax method, we prove the existence of positive solution $u_{\lambda }\in H^1({\mathbb {R}}^3)$ of the above system for $\lambda >0$ large enough. Moreover, we also investigate the concentration behavior of $\{u_{\lambda }\}$ as $\lambda \rightarrow +\infty$.