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

TL;DR

This paper proves the existence of nontrivial solutions for a strongly indefinite nonlinear Choquard equation with critical growth, using variational methods and extending previous results in the field.

Contribution

It introduces new existence results for a class of indefinite Choquard equations with critical exponent, expanding the theoretical understanding of such nonlinear problems.

Findings

Existence of nontrivial solutions established

Solutions found under critical growth conditions

Extension of previous theorems in the literature

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}.