Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains
type: Item
instance of: scholarly article
title: Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains
zbMATH Open document ID: 0701.35068
DOI: 10.1007/BF01168296
author: Donato Passaseo
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155431
review text: The paper is concerned with the following problem: \[ (*)\quad \Delta u+u^{2^*-1}=0\text{ in } \Omega;\quad u>0\text{ in } \Omega;\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with \(n\geq 3\) and \(2^*=2n/(n-2)\) is the critical Sobolev exponent. A well known result of Pohozaev states that the problem has no solution if the domain \(\Omega\) is star-shaped; on the other hand, Bahri and Coron have proved that (*) has at least one solution if \(\Omega\) has ``non trivial'' topology. In this paper it is proved that for every positive integer k there exists a contractible bounded domain \(\Omega\) such that (*) has at least k solutions. This result shows that the existence and the multiplicity of solutions for the problem (*) does not depend on the topology of \(\Omega\) and suggests that the basic point is the topology of suitable nearby domains. The domains \(\Omega\) considered in this paper have a rotational symmetry with respect to one axis, which plays an important role in the proof; but other analogous existence and multiplicity results in domains without any symmetry property are contained in a paper to appear.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4148696
zbMATH Keywords: non trivial topology
zbMATH Keywords: multiplicity
zbMATH Keywords: positive solutions
zbMATH Keywords: contractible domains
zbMATH Keywords: critical Sobolev exponent
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4148696