Ground State Solutions for a Class of Strongly Indefinite Choquard Equations

Abstract

In this paper, we study the existence and concentration of ground state solution for the Choquard equation $$\begin{aligned} \left\{ \begin{array}{ll} \begin{aligned} &{}-\Delta u+V(x)u=\left( \int _{{\mathbb {R}}^N}\frac{A(\epsilon y)|u(y)|^p}{|x-y|^\mu }\mathrm{d}y\right) A(\epsilon x)|u|^{p-2}u~~\text {in}~{\mathbb {R}}^{N},\\ &{}u\in H^1({\mathbb {R}}^N), \end{aligned} \end{array} \right. \end{aligned}$$where $$N\ge 2$$, $$0<\mu <2$$, $$\epsilon $$ is a positive parameter. V is a $${\mathbb {Z}}^N$$-periodic function, and 0 lies in a gap of the spectrum of $$-\Delta + V$$. $$A\in C({\mathbb {R}}^N)$$ satisfies $$\begin{aligned} 0<\inf \limits _{x\in {\mathbb {R}}^N}A(x)\le \lim \limits _{|x|\rightarrow +\infty }A(x) <\sup \limits _{x\in {\mathbb {R}}^N}A(x). \end{aligned}$$