Multiple solutions to the Bahri-Coron problem in some domains with nontrivial topology

TL;DR

This paper demonstrates that in dimensions three and higher, there exist many bounded domains with limited symmetry where the nonlinear elliptic problem admits multiple solutions, including both positive and sign-changing solutions.

Contribution

It shows the existence of multiple solutions to the Bahri-Coron problem in domains with finite symmetry, extending previous results to a broader class of domains.

Findings

Existence of multiple solutions in certain domains

Presence of both positive and sign-changing solutions

Results hold for all dimensions N ≥ 3

Abstract

We show that in every dimension $N\geq3$ there are many bounded domains $\Omega\subset\mathbb{R}^{N}$, having only finite symmetries, in which the Bahri-Coron problem \[-\Delta u=|u| ^{4/(N-2)}u\text{\in}\Omega,\text{\ \}u=0\text{\ on}\partial\Omega, \] has a prescribed number of solutions, one of them being positive and the rest sign changing.