The beginning of the Fučik spectrum for the \(p\)-Laplacian
type: Item
instance of: scholarly article
title: The beginning of the Fučik spectrum for the \(p\)-Laplacian
zbMATH Open document ID: 0947.35068
author: Djairo G. de Figueiredo
author: Mabel Cuesta
author: J.-P. Gossez
published in: Journal of Differential Equations
publication date: 2000-10-30T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\), \(N\geq 1\) and let \(p\) be a real number greater than \(1\). The Fučik spectrum of the \(p\)-Laplacian \(\Delta_p=\operatorname {div}(|\nabla u|^{p-2}\nabla u)\) on \(W^{1,p}_0(\Omega)\) is defined as the set \(\Sigma_p\) of pairs \((\alpha,\beta)\in \mathbb{R}^2\) such that the Dirichlet problem \(-\Delta_pu=\alpha(u^+)^{p-1}-\beta(u^-)^{p-1}\) in \(\Omega\) and \(u=0\) on \(\partial\Omega\) has a nontrivial solution. The usual spectrum of \(-\Delta_p\) corresponds to \(\alpha=\beta\). Clearly, if \(\lambda_1\) is the first eigenvalue of \(-\Delta_p\) on \(W^{1,p}_0\), the set \(\Sigma_p\) contains the two lines \((\lambda_1\times \mathbb{R})\) and \((\mathbb{R}\times\lambda_1)\). The authors prove that these lines are isolated in \(\Sigma_p\). Then, by using the mountain pass theorem, they construct a nontrivial curve in \(\Sigma_p\) and prove that such a curve is in fact the ``first nontrivial curve'' in \(\Sigma_p\). As a consequence, a variational characterization via a mountain pass argument of the second eigenvalue of \(-\Delta_p\) follows. Also, the regularity, monotonicity and asymptotic behaviour of this curve is studied. As an application of the above results, the solvability of the homogeneous Dirichlet problem \(-\Delta_pu=f(x,u)\) in \(\Omega\) is studied assuming that \(f(x,u)/|u|^{p-2}u\) lies asymptotically between \((\lambda_1,\lambda_1)\) and one point \((\alpha,\beta)\) of the first curve in \(\Sigma_p\). To study this problem, the montain pass theorem is used again.
reviewed by: Giovanni Porru
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 1389122
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Fucik spectrum
zbMATH Keywords: nonresonance
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3645
OpenAlex ID: W2065192976
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DOI: 10.1006/JDEQ.1999.3645
description: scientific article; zbMATH DE number 1389122