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Equation de Yamabe sur un ouvert non contractile. (The Yamabe equation on a not contractible domain). (French) Zbl 0629.35041

Let \(\Omega\) be a bounded regular connected not contractible open set on \({\mathbb{R}}^ 3\). Then there exists a solution of \(\Delta u+u^ 5=0\) in \(\Omega\), \(u>0\) in \(\Omega\) and \(u=0\) on \(\partial \Omega\). The proof is obtained showing that \(J: \Sigma\to {\mathbb{R}}\) has a critical point in \(\Sigma_+\), where \(\Sigma =\{u\in H^ 1_ 0(\Omega):| u| =1\}\), \(\Sigma_+=\{u\in \Sigma:\) \(u\geq 0\}\) and \(J(u)=1/\sqrt{\int_{\Omega} u^ 6 dx}\).
Reviewer: G.Bottaro

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J20 Variational methods for second-order elliptic equations
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