Nonlocal Choquard equations involving critical Hardy-Littlewood-Sobolev exponent: the effect of the topology of the domain

TL;DR

This work studies positive solutions to a nonlocal Choquard equation on bounded domains at the upper Hardy-Littlewood-Sobolev exponent, focusing on how the domain's topology influences existence. The authors adapt the Bahri-Coron topological framework to a nonlocal variational setting, employing a reduced functional $J$ built from bubble configurations and deriving a detailed Green's function–based asymptotic expansion. They prove that if the domain has nontrivial homology $H_{k_0}(\Omega)\neq 0$, a positive solution exists, thereby linking topology to solvability in nonlocal critical problems. The results extend topological methods to nonlocal elliptic equations and highlight the role of domain topology in overcoming lack of compactness at criticality.

Abstract

We apply a topological method to prove existence of positive solutions for the nonlineair Choquard equation with upper critical exponent in the sense of Hardy-Littlewood-Sobolev inquality on bounded domains having nontrivial homology group.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4