Schrödinger–Bopp–Podolsky System with Steep Potential Well

Abstract

In this paper, we study the following Schrödinger–Bopp–Podolsky system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda V(x)u+ q^2 \varphi u=f(u)\,\,\\ -\Delta \varphi + a^2\Delta ^2 \varphi =4\pi u^2, \end{array}\right. } \end{aligned}$$

where $u, \varphi :{\mathbb {R}}^{3} \rightarrow {\mathbb {R}}$, $a>0, ~q > 0$, $\lambda$ is real positive parameter, f satisfies supper 2 lines growth. $V \in C({\mathbb {R}}^{3}, {\mathbb {R}})$, which suppose that V(x) repersents a potential well with the bottom $V^{-1}(0)$. There are no results of solutions for the system with steep potential well in the current literature because of the presence of the nonlocal term. Throughout the truncation technique and the parameter-dependent compactness lemma, we get a poistive energy solution $u_{\lambda ,q}$ for $\lambda$ large and q small. In the last part, we explore the asymptotic behavior as $q \rightarrow 0$, $\lambda \rightarrow + \infty$.