On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

TL;DR

This paper investigates the existence of solutions for a nonlinear Choquard equation with a Brezis-Nirenberg type critical exponent, extending the understanding of such equations in bounded domains with Lipschitz boundaries.

Contribution

The paper establishes new existence results for a critical nonlinear Choquard equation involving the Hardy-Littlewood-Sobolev critical exponent in bounded domains.

Findings

Existence of solutions for certain parameter ranges.

Extension of Brezis-Nirenberg results to nonlocal Choquard equations.

Analysis of critical exponent cases in bounded domains.

Abstract

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda u\4.14mm\mbox{in}\1.14mm \Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $\lambda$ is a real parameter, $N\geq3$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.