\(p\)-Laplacian problems involving critical Hardy-Sobolev exponents
type: Item
instance of: scholarly article
title: \(p\)-Laplacian problems involving critical Hardy-Sobolev exponents
zbMATH Open document ID: 1401.35109
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2018-07-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1609.01804
review text: The authors study the nonlinear eigenvalue problem \[ -\Delta_p u=\mu|u|^{p-2} u+{|u|^{p^*(s)-2}u\over |x|^s}\quad\text{in }\Omega, \] \(u=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(1<p<N\), \(0<s<p\) and \(p^*(s)= (N-s)p/(N-p)\) is the critical Hardy-Sobolev exponent. The authors prove existence, multiplicity and bifurcation results extending to \(\mu\geq\mu_1\) (where \(\mu_1\) is the first eigenvalue of \(-\Delta_p\)) results which are known for \(\mu<\mu_1\). The methods of proof include mainly several versions of variational arguments, in particular some non-standard linking theorems involving sub-level sets due to Perera and Szulkin. An alternative sequence of eigenvalues for the \(p\)-Laplacian obtained via a cohomological index is also used.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6909894
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: critical Hardy-Sobolev exponent
zbMATH Keywords: multiplicity and bifurcation results
author: Kanishka Perera
author: Wenming Zou
reviewed by: Jesús Hernández
MaRDI profile type: Publication
OpenAlex ID: W2963433463
arXiv ID: 1609.01804
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DOI: 10.1007/S00030-018-0517-7
description: scientific article