Critical point theory for indefinite functionals with symmetries
type: Item
instance of: scholarly article
title: Critical point theory for indefinite functionals with symmetries
zbMATH Open document ID: 0853.58027
author: Thomas J. Bartsch
author: Mónica Clapp
published in: Journal of Functional Analysis
publication date: 1997-01-05T00:00:00Z
review text: Let \(X\) be a Hilbert space and \(G\) a compact Lie group acting orthogonally on \(X\). Let \(\phi \in C^1(X,\mathbb{R})\) be a strongly indefinite functional, invariant with respect to the action of \(G\). In order to find critical points of \(\phi\), the authors introduce an equivariant version of the limit relative category of \textit{G. Fournier, D. Lupo, M. Ramos}, and \textit{M. Willem} [Limit relative category and critical point theory, Dynamics Rep. 3, 1-24 (1994; Zbl 0794.58007)]. This category is employed in order to obtain generalizations of the symmetric mountain pass theorem and the symmetric linking theorem. In the mountain pass case it is shown that \(\phi\) has an unbounded sequence of critical values. The hypotheses made here are weaker than the usual ones, even in the case of \(G = Z_2\). In the linking theorem a lower bound for the number of critical orbits is given in terms of the asymptotic behaviour of \(\phi\) at 0 and \(\infty\). The proofs are obtained by reduction to equivariant (relative) Borsuk-Ulam theorems. An application of the mountain pass theorem is given to the elliptic system \(\Delta v = F_v(x,v,w)\), \(-\Delta w = F_w(x,v,w)\) in \(\Omega\), \(v = w = 0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(F\) is superquadratic (in a weaker sense than the usual). It is shown that if \(F\) is even in \(w\) or in \((v,w)\), then this system has infinitely many solutions. An application of the linking theorem is also sketched. The problem considered is that of existence of periodic solutions to an asymptotically linear Hamiltonian system with symmetric Hamiltonian.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E30
zbMATH DE Number: 889770
zbMATH Keywords: strongly indefinite functional
zbMATH Keywords: action of Lie group
zbMATH Keywords: limit relative category
zbMATH Keywords: Borsuk-Ulam theorems
zbMATH Keywords: semilinear elliptic system
zbMATH Keywords: Hamiltonian system
MaRDI profile type: Publication
full work available at URL: jfan.1996.0058
OpenAlex ID: W2041664892
DOI: 10.1006/JFAN.1996.0058
description: scientific article; zbMATH DE number 889770