The topological impact of critical points at infinity in a variational problem with lack of compactness: The dimension 3
There is investigated the nonlinear elliptic problem \[ -\Delta u=u^p,\quad u>0 \text{ in } \Omega, \] \[ u=0 \text{ on } \partial\Omega, \] where \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^N, N\geq 3\), and \(p=\frac{N+2}{N-2}\) is the critical Sobolev exponent. There is studied the existence or nonexistence of solutions to the mentioned problem in dependence on the topological structure of the domain \(\Omega\). A result of \textit{A. Bahri, Y. Li} and \textit{O. Rey} [Calc. Var. Partial Differ. Equ. 3, No.~1, 67-93 (1995; Zbl 0814.35032)] previously known in higher dimensions and concerning the topological effect of critical points at infinity on the level sets of functionals associated to an elliptic problem with critical nonlinearity, is extended to dimension 3.
scientific article; zbMATH DE number 1493605