Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains

TL;DR

This paper investigates the existence and number of positive solutions for a slightly subcritical Choquard problem on bounded domains, linking the solutions count to the domain's topological complexity.

Contribution

It establishes a relationship between the number of positive solutions and the Lusternik-Schnirelmann category of the domain for a subcritical Choquard problem.

Findings

Number of positive solutions depends on domain topology.

Existence of at least cat(D)+1 solutions near critical exponent.

Solutions count increases with domain complexity.

Abstract

In this paper we study a slightly subcritical Choquard problem on a bounded domain D. We prove that the number of positive solutions depends on the topology of the domain. In particular when the exponent of the nonlinearity approaches the critical one, we show the existence of cat(D)+1 solutions. Here cat(D) denotes the Lusternik-Schnirelmann category.