Existence and concentration of ground state solutions for doubly critical Schrödinger–Poisson-type systems

Abstract

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrodinger–Poisson-type system with doubly critical growth $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u-\phi |u|^3u=|u|^4u+f(u),&{}\mathrm{in}\; \mathbb {R}^3,\\ -\varepsilon ^2\Delta \phi =|u|^5,&{}\mathrm{in}\; \mathbb {R}^3, \end{array}\right. } \end{aligned}$$ where $$\varepsilon >0$$ is a small parameter. By employing the concentration-compactness principle and mountain pass theorem, we prove the existence of positive ground state solutions $$v_\varepsilon $$ with exponential decay at infinity for $$\varepsilon $$ sufficiently small under some suitable assumptions on the potential V and nonlinearity f. Moreover, as $$\varepsilon \rightarrow 0^+$$ , $$v_\varepsilon $$ concentrates around a global minimum point of V.