Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains
type: Item
instance of: scholarly article
title: Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains
zbMATH Open document ID: 1080.35058
author: Alexandru Kristály
published in: Set-Valued Analysis
publication date: 2005-06-15T00:00:00Z
review text: Let \(\Omega=\omega\times \mathbb R^{N-m}\), where \(\omega\) is a bounded open set in \(\mathbb R^m\), \(m\geq 1\), \(N-m\geq 2\). Assume that \(F:\Omega\times \mathbb R\rightarrow \mathbb R\) is a Carathéodory function with subcritical growth. This paper is devoted to the study of the following eigenvalue problem for hemivariational inequalities: find \(u\in H^1_0(\Omega)\) such that \(\int_\Omega\nabla u\nabla w\,dx+\lambda\int_\Omega F^0(x,u(x);-w(x))\,dx\geq 0\) for all \(w\in H^1_0(\Omega)\), where \(F^0(x,s;t)\) denotes the Clarke generalized directional derivative of the locally Lipschitz mapping \(F(x,\cdot )\) at the point \(s\in \mathbb R\) in the direction \(t\in\mathbb R\). The first main result of the paper establishes the existence of at least three distinct, axially symmetric solutions for certain eigenvalues. Next, applying the fountain theorem, the author proves the existence of infinitely many axially symmetric solutions, but also of axially nonsymmetric solutions in certain dimensions. In both cases the principle of symmetric criticality plays a crucial role in proofs. The paper is carefully written and the reviewer believes that the methods developed in the paper can be applied to other classes of nonlinear eigenvalue problems.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2176209
zbMATH Keywords: strip-like domain
zbMATH Keywords: hemivariational inequalities
zbMATH Keywords: principle of symmetric criticality
zbMATH Keywords: Clarke generalized derivative
zbMATH Keywords: existence
zbMATH Keywords: symmetric solutions
zbMATH Keywords: axially nonsymmetric solutions
MaRDI profile type: Publication
full work available at URL: s11228-004-6565-7
OpenAlex ID: W2167195226
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DOI: 10.1007/S11228-004-6565-7
description: scientific article; zbMATH DE number 2176209