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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
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