Bahri, Abbas; Coron, Jean-Michel Equation de Yamabe sur un ouvert non contractile. (The Yamabe equation on a not contractible domain). (French) Zbl 0629.35041 Rend. Ist. Mat. Univ. Trieste 18, 1-15 (1986). 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 Cited in 2 Documents 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 Keywords:Yamabe equation; existence; not contractible open set; critical point PDFBibTeX XMLCite \textit{A. Bahri} and \textit{J.-M. Coron}, Rend. Ist. Mat. Univ. Trieste 18, 1--15 (1986; Zbl 0629.35041)