Computations of critical groups and applications to asymptotically linear wave equation and beam equation
type: Item
instance of: scholarly article
title: Computations of critical groups and applications to asymptotically linear wave equation and beam equation
zbMATH Open document ID: 1073.35147
author: Pingan Zeng
author: Yuxia Guo
author: Jia-Quan Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-02-23T00:00:00Z
review text: The use of critical groups to study the existence and multiplicity of solutions to partial differential equations arising from the application of a variational principle relies heavily on Morse theory. However, there is a problem in the case of strongly indefinite functionals. A new theory is developed for the Morse index for strongly indefinite functionals. This theoretical development constitutes some half of the paper. Two problems, the asymptotically linear wave equation and the asymptotically linear beam equation, to which class of equations the theoretical development is directed, are given. For the former nine cases for which at least one nontrivial weak solution exists are identified. With an additional consumption, the existence of a nonzero nondegenerate periodic solution, a further nonzero solution occurs. Similar results are found for the latter which is observed to be very similar to the former under the conditions stated.
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 2137692
zbMATH Keywords: wave equation
zbMATH Keywords: beam equation
zbMATH Keywords: asymptotically linear wave equation
zbMATH Keywords: Morse theory
zbMATH Keywords: nondegenerate periodic solution
reviewed by: P. G. L. Leach
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.06.029
OpenAlex ID: W2037331379
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DOI: 10.1016/J.JMAA.2004.06.029
description: scientific article; zbMATH DE number 2137692