Infinitely many bound state solutions of Choquard equations with potentials

Abstract

In this paper, we consider the following Choquard equation $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \Delta u+a(x)u=(I_\alpha *|u|^p)|u|^{p-2}u, &{}x\in {\mathbb {R}}^N,\\ u(x)\rightarrow 0,&{}|x|\rightarrow +\infty , \end{array}\right. {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\hbox {CH})} \end{aligned}$$ where $$N\ge 3, I_\alpha $$ is a Riesz potential, $$\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}$$ and a(x) is a given nonnegative potential function. Under some assumptions of asymptotic properties on a(x) at infinity and according to a concentration compactness argument, we obtain infinitely many solutions of (CH), whose energy can be arbitrarily large.