Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential
type: Item
instance of: scholarly article
title: Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential
zbMATH Open document ID: 1221.35269
author: Mihai Mihăilescu
author: Vicenţiu D. Rădulescu
published in: Journal d'Analyse Mathématique
publication date: 2011-03-10T00:00:00Z
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
review text: This paper deals with nonlinear eigenvalue problems of the type \(Au=\mu Bu\), generalizing problems with the usual \(p\)-Laplacian for \(p> 1\) constant. Here \(A\) (resp. \(B\)) is a differential operator( resp. a zero order term ) which is a sum of two \(p\)-Laplacians with different \(p\)'s ( resp. two zero order terms with different powers). Now \(p(x)> 0\) is a non-constant function. These problems have been studied in the last years in the frame of variable exponent Lebesgue spaces. Moreover, the operator A may include a zero order term where an indefinite changing sign weight \(V(x)\) arises. Under suitable assumptions it is proved that for given \(V\), an eigenvalue \(E(V)\) exists such that any \(\mu> E(V)\) is an eigenvalue and any \(\mu< E(V)\) is not an eigenvalue. It is also proved that \(E(V)\) can be minimized on closed, convex, bounded subsets in a suitable space. Since the problem has not a variational character, most of the usual arguments should be modified or replaced by different ones.
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5864342
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: variable exponent Lebesgue space
zbMATH Keywords: sign-chaning potential
zbMATH Keywords: optimization of eigenvalues
zbMATH Keywords: nonhomogeneous problem
reviewed by: Jesús Hernández
MaRDI profile type: Publication
OpenAlex ID: W1993097417
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DOI: 10.1007/S11854-010-0018-Z
description: scientific article; zbMATH DE number 5864342