A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality

Abstract

In this paper we are concerned with the following nonlinear Choquard equation $$-\Delta u+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $N\geq4$, $0<\mu<N$ and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. If $0$ lies in a gap of the spectrum of $-\Delta +V$ and $g(x,u)$ is of critical growth due to the Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in \cite{AC, KS, MS2}.