Structure of solutions set of nonlinear eigenvalue problems
type: Item
instance of: scholarly article
title: Structure of solutions set of nonlinear eigenvalue problems
zbMATH Open document ID: 1344.47042
author: Xian Xu
author: Jingxian Sun
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-12-21T00:00:00Z
review text: The authors use degree theory and global bifurcation arguments in order to obtain information concerning the structure of the solution set for some nonlinear eigenvalue problems. This work goes back until \textit{P. H. Rabinowitz}'s global bifurcation theorems giving the alternative between unbounded continua of solutions and closed loops [J. Funct. Anal. 7, 487--513 (1971; Zbl 0212.16504)]. Some general results in both directions are proved here by using degree theory and set theoretic topology theorems. An application is given to the a problem \(-\Delta u=\mu m(x)u+ b(x)u^a\) in \(\Omega\), \(\partial u/\partial n= 0\) on \(\partial\Omega\), where \(m\) and \(b\) change sign on \(\Omega\), \(m\) changes sign on \([x\in\Omega: b(x)>0]\) and also \(1< a<(N+2)/(N-2)\). In particular, the authors prove the existence of loops improving previous work by \textit{K. J. Brown} [J. Differ. Equations 239, No. 2, 296--310 (2007; Zbl 1331.35129)].
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 6522721
zbMATH Keywords: global bifurcation
zbMATH Keywords: degree theory
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: nonlinear elliptic problem
zbMATH Keywords: loop of solutions
zbMATH Keywords: connected component
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.11.024
OpenAlex ID: W2181151802
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DOI: 10.1016/J.JMAA.2015.11.024
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description: scientific article; zbMATH DE number 6522721