## Elliptic equations with limiting Sobolev exponents - The impact of topology.(English)Zbl 0601.35043

We discuss a number of recent results concerning the existence of a solution for the problem $$-\Delta u=u^ p+a(x)u$$ on $$\Omega$$, $$u>0$$ on $$\Omega$$, and $$u=0$$ on $$\partial \Omega$$, where $$\Omega \subset {\mathbb{R}}^ n$$ is a smooth bounded domain, $$p=(N+2)/(N-2)$$ and a(x) is a given function.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

### Keywords:

limiting Sobolev exponents; topology; existence