The effect of the domain shape on the existence of positive solutions of the equation \(\Delta u + u^{2^*-1}= 0\)
type: Item
instance of: scholarly article
title: The effect of the domain shape on the existence of positive solutions of the equation \(\Delta u + u^{2^*-1}= 0\)
zbMATH Open document ID: 0812.35032
DOI: 10.12775/TMNA.1994.002
author: Donato Passaeo
published in: Topological Methods in Nonlinear Analysis
publication date: 1994-10-18T00:00:00Z
review text: The author considers his work on the existence and multiplicity of positive solutions of the Dirichlet problem for \(-\Delta u= u^ p\) in \(\Omega\) where \(p\) is the critical exponent. He obtains here some interesting existence results for certain small perturbations of \(\Omega\) which change the topology of \(\Omega\). He also obtains some multiplicity results. It is interesting that the solutions he obtains are quite different to the ones obtained by domain variation methods [as in the reviewer's paper, \textit{E. N. Dancer}, Bull. Lond. Math. Soc. 20, No. 6, 600-602 (1988; Zbl 0662.35043)], when they can be applied. Sometimes, this enables one to improve the multiplicity results. Reviewer's remark. The non-existence results of Carpio, Comte and Levandonskii mentioned in the introduction becomes less surprising when one realizes that their domains are conformally equivalent to star shaped domains.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 640268
zbMATH Keywords: critical exponent
zbMATH Keywords: multiplicity results
zbMATH Keywords: domain variation methods
reviewed by: E. N. Dancer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 640268