Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size

Abstract

We consider the problem $-\Delta u = \left\vert u\right\vert ^{2^\ast-2} u\,{\rm in}\,\Omega, \quad u = 0\,{\rm on}\,\partial\Omega,$ where Ω is a bounded smooth domain in $\mathbb{R}^{N}$, N ≥ q3, and $2^{\ast}=\frac{2N}{N-2}$ is the critical Sobolev exponent. We assume that Ω is annular shaped, i.e. there are constants R ₂ >  R ₁ >  0 such that $\{x \in \mathbb{R}^{N} : R_{1} < |x| < R_{2}\} \subset \Omega$ and $0 \not\in \Omega.$ We also assume that Ω is invariant under a group Γ of orthogonal transformations of $\mathbb{R}^{N}$ without fixed points. We establish the existence of multiple sign changing solutions if, either Γ is arbitrary and R ₁/R ₂ is small enough, or R ₁/R ₂ is arbitrary and the minimal Γ-orbit of Ω is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Möbius transformations.