On the dependence on \(p\) of the variational eigenvalues of the \(p\)-Laplace operator
type: Item
instance of: scholarly article
title: On the dependence on \(p\) of the variational eigenvalues of the \(p\)-Laplace operator
zbMATH Open document ID: 1334.35196
author: Marco Degiovanni
author: Marco Marzocchi
published in: Potential Analysis
publication date: 2016-01-25T00:00:00Z
review text: It is known that the quasilinear eigenvalue problem \(-\Delta_p u-\mu|u|^{p-2}u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(p>1\), and \(\Delta\) is the quasilinear differential operator called \(p\)-Laplacian, has a first positive eigenvalue \(\mu^p>0\), which is simple, having a positive eigenfunction \(u\in W^{1,p}(\Omega)\) (normalized by \(\| u\|_p=1\)). It is also known that \(\mu_1\) as a function of \(p\) is right-continuous, while some condition on \(\partial\Omega\) is required for left-continuity. In this case, we have \[ \lim_{s\to p^-}\mu^s_1= \mu^p_1\tag{1} \] and \[ \lim_{s\to p^-}\int\nabla u_s-\nabla u_p|^p\,dx= 0.\tag{2} \] It was proved by Lindqvist that (2) implies (1), the converse being an open problem. Here, the authors introduce a bigger space and an associated eigenvalue problem as a tool in order to prove (Theorem 4.1) that five other statements are equivalent to (1). In particular, this allows to settle the open problem in the affirmative.
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 6535136
zbMATH Keywords: nonlinear eigenvalues
zbMATH Keywords: \(p\)-Laplace operator
zbMATH Keywords: variational convergence
zbMATH Keywords: continuous dependence of eigenvalues
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: s11118-015-9487-0
OpenAlex ID: W1691461481
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DOI: 10.1007/S11118-015-9487-0
description: scientific article; zbMATH DE number 6535136