Existence and concentration behavior of solutions for the logarithmic Schrödinger–Bopp–Podolsky system

Abstract

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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u-\phi u=u \log u^2,&{} \text {in}~{\mathbb {R}}^{3},\\ -\varepsilon ^2\Delta \phi +\varepsilon ^4\Delta ^2\phi =4\pi u^2,~~~~~~~~~~~&{}\text {in}~{\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}$$

where $\varepsilon$ is a small positive parameter and $V(x)\in C({\mathbb {R}}^3,{\mathbb {R}})$. Under the global condition on potential V(x), we prove the existence of positive solution $u_{\varepsilon }\in H^1({\mathbb {R}}^3)$ of above system for $\varepsilon >0$ small enough by applying the Variational Methods developed by Szulkin for the functional which is a sum of a $C^1$ functional with a convex lower semicontinuous functional. Moreover, we also investigate the concentration behavior of $\{u_{\varepsilon }\}$ as $\varepsilon \rightarrow 0$.