The topological impact of critical points at infinity in a variational problem with lack of compactness: The dimension 3
type: Item
instance of: scholarly article
title: The topological impact of critical points at infinity in a variational problem with lack of compactness: The dimension 3
zbMATH Open document ID: 0952.35051
author: Olivier Rey
published in: Advances in Differential Equations
publication date: 2000-08-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1493605
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: critical nonlinearity
zbMATH Keywords: critical points at infinity
zbMATH Keywords: Morse theory
reviewed by: A. E. Shishkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1493605