On a class of hemivariational inequalities at resonance.
type: Item
instance of: scholarly article
title: On a class of hemivariational inequalities at resonance.
zbMATH Open document ID: 1041.49007
author: Nikolaos Halidias
author: Z. Naniewicz
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-15T00:00:00Z
review text: The paper studies a Dirichlet boundary value problem involving the \(p\)-Laplacian \(\Delta_p\) and a locally Lipschitz potential which is resonant at the first eigenvalue of \(-\Delta_p\). This type of problems is called resonant hemivariational inequalities. A specific feature of the paper is that it is only assumed a unilateral growth condition for the generalized directional derivative of the locally Lipschitz potential. The main result of the paper establishes the existence of a solution in an appropriate weak sense. The approach is variational, but not relying directly on the nonsmooth critical point theory. Specifically, due to the generality of the imposed growth condition, the functional associated to the problem is not locally Lipschitz, so it is not possible to apply readily the critical point theory for locally Lipschitz functionals. To overcome this difficulty the authors develop a finite-dimensional approximation, apply on each finite dimensional space a version of the mountain pass theorem for locally Lipschitz functions to obtain approximate solutions and pass to the limit using the Dunford-Pettis criterion.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 2057710
zbMATH Keywords: hemivariational inequality
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: resonance
zbMATH Keywords: critical point theory
zbMATH Keywords: locally Lipschitz functional
zbMATH Keywords: noncoercive unilateral growth condition
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.08.044
OpenAlex ID: W2044657916
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DOI: 10.1016/J.JMAA.2003.08.044
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description: scientific article; zbMATH DE number 2057710