Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

TL;DR

This paper establishes the existence of multiple positive solutions for a Hardy-Littlewood-Sobolev critical elliptic problem in non-contractible annular domains, using variational methods and topological tools when the domain's hole is small.

Contribution

It demonstrates the existence of four positive solutions to a nonlocal elliptic problem with critical exponent in annular domains, employing Lusternik-Schnirelmann theory.

Findings

Proved existence of four positive solutions in annular domains.

Applied variational methods and topological tools for solution multiplicity.

Results hold when the inner hole of the annulus is sufficiently small.

Abstract

The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u+f \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded annular domain in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=\frac{2N-\mu}{N-2}$, $f \in L^{\infty}(\Om)$ and $f \geq 0$. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of the annulus is sufficiently small.