## Topologie et cas limite des injections de Sobolev. (Topology and limit case for Sobolev imbeddings).(French)Zbl 0569.35032

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

### 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