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

Abstract

We show that in every dimension $N\ge 3$ there are many bounded domains $\Omega \subset {\mathbb{R}}^N$, having only finite symmetries, in which the Bahri-Coron problem \[-Δu=|u| ^{4/(N-2)}u\text{\in}Ω,\text{\ \}u=0\text{\ on}\partialΩ, \] has a prescribed number of solutions, one of them being positive and the rest sign changing.

To appear in Proceedings of the American Mathematical Society