Some existence results for superlinear elliptic boundary value problems involving critical exponents
type: Item
instance of: scholarly article
title: Some existence results for superlinear elliptic boundary value problems involving critical exponents
zbMATH Open document ID: 0614.35035
DOI: 10.1016/0022-1236(86)90094-7
author: Sergio Solimini
author: Giovanna Cerami
author: Michael Struwe
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with the superlinear elliptic boundary value problem \[ - \Delta u-\lambda u=u | u|^{2^*-2}\quad in\quad \Omega;\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is a smoothly bounded domain in \(R^ n\), \(n>2\), \(\lambda\in R\), \(2^*=2n/(n-2)\) and \(2^*\) is the limiting Sobolev exponent for the embedding \(H^ 1_ 0(\Omega)\to L^ p(\Omega)\). Solving this problem is equivalent to finding critical points in \(H^ 1_ 0(\Omega)\) of the energy functional \[ I_{\lambda}(u)=(1/2)\int_{\Omega}(| \nabla u|^ 2-\lambda | u|^ 2)dx-(1/2)\int_{\Omega}| u|^{2^*} dx. \] First, based on the global compactness theorem for the problem (1), the compactness question is discussed. Then some results about the existence and multiplicity of solutions to the above problem are obtained.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3994107
zbMATH Keywords: superlinear elliptic boundary value problem
zbMATH Keywords: limiting Sobolev exponent
zbMATH Keywords: critical points
zbMATH Keywords: energy functional
zbMATH Keywords: global compactness
zbMATH Keywords: existence
zbMATH Keywords: multiplicity of solutions
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90094-7
OpenAlex ID: W1994484892
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description: scientific article; zbMATH DE number 3994107