On the Fučik spectrum of Hardy-Sobolev operator
type: Item
instance of: scholarly article
title: On the Fučik spectrum of Hardy-Sobolev operator
zbMATH Open document ID: 1157.35446
DOI: 10.1016/S0362-546X(01)00886-0
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-11-28T00:00:00Z
review text: The author deals with the Fučik spectrum of the Hardy-Sobolev operator by \(L_\mu\) on \(W^{1,p}_0(\Omega)\) defined as the set \(\Sigma_{p,\mu}\) of these \((\alpha,\beta)\in\mathbb{R}^2\) such that \[ \begin{gathered} L_\mu u:= -\Delta_pu- {\mu\over|x|^p}|u|^{p- 2}u= \alpha(u^+)^{p-1}- \beta(u^-)^{p-1}\quad\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega\end{gathered} \] has a nontrivial solution \(u\in W^{1,p}_0(\Omega)\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) containing \(0\), \(N\geq 2\), \(1< p< n\), \(\Delta_p\) is the \(p\)-Laplacian, \(\mu<({n-p\over p})^p\). Using comparison theorem the author shows that the first eigenvalue \(\mu_1\) is simple and isolated in the spectrum \(L_\mu\). Moreover, the author constructs a nontrivial curve \(C\) in \(\Sigma_{p,\mu}\), that is so-called the ``first curve'' and proves that the two trivial lines \(\mu_1\times\mathbb{R}\) and \(\mathbb{R}\times \mu_1\) are isolated in \(\Sigma_{p,\mu}\).
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1836638
zbMATH Keywords: Fučik spectrum
zbMATH Keywords: Hardy-Sobolev operator
zbMATH Keywords: eigenvalue
author: K. Sreenadh
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836638