On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents

TL;DR

This paper investigates the existence and multiplicity of solutions for a nonlinear nonlocal Choquard equation with critical exponents, using variational methods under various nonlinearities.

Contribution

It introduces new existence and multiplicity results for Choquard equations with Hardy-Littlewood-Sobolev critical exponents, expanding the understanding of such nonlocal problems.

Findings

Established existence of solutions under certain conditions.

Proved multiplicity of solutions for specific nonlinearities.

Applied variational methods to analyze the problem.

Abstract

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, $$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu<N$ and $2_{\mu}^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.