Coron problem for nonlocal equations invloving Choquard nonlinearity

TL;DR

This paper investigates a nonlocal elliptic PDE with Choquard nonlinearity, proving the existence of positive solutions in annular domains with small inner holes, expanding understanding of such equations in bounded domains.

Contribution

It establishes the existence of positive solutions for a class of nonlocal equations with Choquard nonlinearity in annular domains with small inner holes.

Findings

Existence of positive solutions in annular domains.

Solutions depend on the size of the inner hole.

Method applicable to similar nonlocal problems.

Abstract

We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=\frac{2N-\mu}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.