Coron, Jean-Michel Topologie et cas limite des injections de Sobolev. (Topology and limit case for Sobolev imbeddings). (French) Zbl 0569.35032 C. R. Acad. Sci., Paris, Sér. I 299, 209-212 (1984). Let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^ n\) such that for some \(\lambda >1\), some \(x_ 0\in {\mathbb{R}}^ n\) and some \(R_ 1,R_ 2>0\) with \(R_ 2>\lambda R_ 1\), we have \(\{x\in {\mathbb{R}}^ n:\quad R_ 1\leq | x-x_ 0| \leq R_ 2\}\subset \Omega\) and \(x\in {\mathbb{R}}^ n:\quad | x-x_ 0| <R_ 1\}\not\subset {\bar \Omega}.\) The author proves that there is at least one solution of the problem \[ -\Delta u=u^{(n+2)/(n-2)},\quad u>0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega. \] Reviewer: D.E.Edmunds Cited in 3 ReviewsCited in 143 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35A30 Geometric theory, characteristics, transformations in context of PDEs Keywords:existence; Sobolev embedding; limiting case PDF BibTeX XML Cite \textit{J.-M. Coron}, C. R. Acad. Sci., Paris, Sér. I 299, 209--212 (1984; Zbl 0569.35032) OpenURL