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

Clapp, Mónica; Faya, Jorge

doi:10.1090/S0002-9939-2013-12043-5

Multiple solutions to the Bahri-Coron problem in some domains with nontrivial topology
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Proc. Amer. Math. Soc. 141 (2013), 4339-4344 Request permission

Abstract:

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

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