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

Abstract

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