On the normalized eigenvalue problems for nonlinear elliptic operators
type: Item
instance of: scholarly article
title: On the normalized eigenvalue problems for nonlinear elliptic operators
zbMATH Open document ID: 1112.47051
author: Jing Lin
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-26T00:00:00Z
review text: Let \(D\) be a bounded open in a real Banach space \(X\). This paper is devoted to the study of the nonlinear eigenvalue problem \(Au-C(\lambda ,u)=0\) in \(D\), \(u=0\) on \(\partial D\), where \(\lambda\geq 0\), \(A\) is an unbounded maximal monotone operator on \(X\), and \(C\) is defined and continuous on \(\mathbb R_+\times\partial D\) such that zero is not in the weak closure of a subset of \(\{C(\lambda ,u)/\| C(\lambda ,u)\| \}\). The main result of the present paper establishes that, under some natural assumptions, this nonlinear eigenvalue problem does not depend on any properties of \(C\) located in \(\mathbb R_+\times D\). The main idea of the proof consists in the construction of a continuous extension of \(C\) based on its values on \(\overline{\mathbb R_+}\times\partial D\) such that its new range inherits the weak nonconvexity property. An application to nonlinear elliptic equations under degenerate conditions is provided in the last section of the paper.
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 5129632
zbMATH Keywords: weak closure
zbMATH Keywords: continuous extension
zbMATH Keywords: maximal monotone
zbMATH Keywords: eigenvalue
zbMATH Keywords: nonlinear elliptic equation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.06.072
OpenAlex ID: W4238507970
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DOI: 10.1016/J.JMAA.2006.06.072
description: scientific article; zbMATH DE number 5129632