Bifurcation theory for symmetric potential operators and the equivariant cup-length
type: Item
instance of: scholarly article
title: Bifurcation theory for symmetric potential operators and the equivariant cup-length
zbMATH Open document ID: 0682.58037
DOI: 10.1007/BF02570878
author: Thomas J. Bartsch
author: Mónica Clapp
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183775
review text: Let G be a compact Lie group, E a Hilbert space on which G acts orthogonally and \(\phi: {\mathbb{R}}\times E\to {\mathbb{R}}\) a \(C^ 2\)-functional which is invariant with respect to the action of G on E. Assume that \(0\in E\) is a critical point of \(\phi(\lambda\),.) for all \(\lambda\in {\mathbb{R}}\). We are interested in critical points of \(\phi(\lambda\),.) for \(\lambda\) near \(\lambda_ 0\) which bifurcate from \((\lambda_ 0,0)\). We give lower bounds for the number of bifurcating G-orbits of critical points. For \(G={\mathbb{Z}}/2\) this problem has been considered e.g. by \textit{E. R. Fadell} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 26, 48-67 (1977; Zbl 0363.47029)]. We generalize their result in that we allow more general potentials and by replacing \({\mathbb{Z}}/2\) by an arbitrary compact Lie group. The main result about bifurcation is reduced to a generalized mountain pass theorem. The proof uses a new index theory based on the concept of cup length in the equivariant context.
reviewed by: Thomas J. Bartsch
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 4117518
zbMATH Keywords: critical points
zbMATH Keywords: bifurcation
zbMATH Keywords: generalized mountain pass theorem
MaRDI profile type: Publication
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Wikidata QID: Q126270986
description: scientific article; zbMATH DE number 4117518