On the Fučík spectrum of the \(p\)-Laplacian with no-flux boundary condition
type: Item
instance of: scholarly article
title: On the Fučík spectrum of the \(p\)-Laplacian with no-flux boundary condition
zbMATH Open document ID: 1503.35083
author: Giuseppina D'Aguì
author: Angela Sciammetta
author: Patrick Winkert
published in: Nonlinear Analysis. Real World Applications
publication date: 2022-11-11T00:00:00Z
review text: This article discusses the quasilinear problem \(-\Delta_p u=a(u^+)^{p-1}-b(u^-)^{p-1}\) in \(\Omega\) where \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^N\), \(N\geq 2\). On the boundary the solution \(u\) is assumed to fulfill \(u=\mathrm{constant}\) and \(\int_{\partial\Omega} |\nabla u|^{p-2}\nabla u\cdot \nu d\sigma=0\). In the above, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplace operator, \(1<p<\infty\), \(\nu\) denotes the outer unit normal to \(\partial\Omega\), \(u^{\pm}=\max\{\pm u, 0\}\). The authors investigate the Fučík spectrum, namely the set of all pairs \((a,b)\) for which the above problem has nontrivial solutions. It is obtained that this spectrum has a first nontrivial curve \(\mathcal{C}\) which is Lipschitz, decreasing and with a certain asymptotic behavior. Some characterizations of the second eigenvalue \(\lambda_2\) of the \(p\)-Laplace operator are obtained.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 7614824
zbMATH Keywords: \(p\)-Laplace operator
zbMATH Keywords: eigenvalues problem
zbMATH Keywords: first nontrivial curve
zbMATH Keywords: Fučík spectrum
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2022.103736
OpenAlex ID: W4292116828
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DOI: 10.1016/J.NONRWA.2022.103736
description: scientific article; zbMATH DE number 7614824