Branching points for a class of variational operators
type: Item
instance of: scholarly article
title: Branching points for a class of variational operators
zbMATH Open document ID: 0931.47051
DOI: 10.1007/BF02786940
author: A. Ambrosetti
published in: Journal d'Analyse Mathématique
publication date: 2000-02-16T00:00:00Z
review text: Bifurcation problems of the type \[ Lu + H(u) = \lambda u \] in a Hilbert space \(E\) are studied. Here \(L:E \to E\) is a linear symmetric Fredholm operator, \(H(u) = h'(u)\), \(h\) is a functional on \(E\), \(h(0)=h'(0)=h''(0)=0, \lambda\) is a real parameter. It is proved that for a large class of functionals, any eigenvalue \(\mu\) of odd multiplicity is a branching point in the sense that the closure \(\Sigma\) of the set of nontrivial solutions contains a connected set \(S\) such that \([\mu,0] \in S, \;S \setminus\{[\mu,0]\} \neq \emptyset \). (Let us recall that known counterexamples show that in general, \(\Sigma\) can contain only isolated points.) Moreover, under natural assumptions about the higher derivatives of \(h\) at zero, it is proved that the set \(\Pi S = \{\lambda \in R;\;[\lambda,u] \in \Sigma \text{ for some } u\}\) contains an interval around \(\mu\) or a one sided neighbourhood \(U\) of \(\mu\). Further, for any \(\mu \in U\) there are at least two nontrivial solutions, and in the case of a multiple eigenvalue at least two couples of nontrivial solutions on \(S\). Examples of boundary value problems for PDE are given.
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 58E07
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 1274834
zbMATH Keywords: bifurcation
zbMATH Keywords: branches of nontrivial solutions
zbMATH Keywords: variational operators
zbMATH Keywords: linear symmetric Fredholm operator
reviewed by: Milan Kučera
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02786940
OpenAlex ID: W2042329336
description: scientific article; zbMATH DE number 1274834