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Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size. (English) Zbl 1143.35052

Summary: We consider the problem \[ -\Delta u = \left| u\right|^{2^\ast-2} u\,{\text{in}}\,\Omega, \quad u = 0\,{\text{on}}\,\partial\Omega, \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(N \geq 3\), and \(2^{\ast}=\frac{2N}{N-2}\) is the critical Sobolev exponent. We assume that \(\Omega\) 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 \(\Omega\) is invariant under a group \(\Gamma\) of orthogonal transformations of \(\mathbb{R}^{N}\) without fixed points. We establish the existence of multiple sign changing solutions if, either \(\Gamma\) is arbitrary and \(R_{1}/ R_{2}\) is small enough, or \(R_{1}/ R_{2}\) is arbitrary and the minimal \(\Gamma\)-orbit of \(\Omega\) 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.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B33 Critical exponents in context of PDEs
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