Bifurcation points of a degenerate elliptic boundary-value problem
type: Item
instance of: scholarly article
title: Bifurcation points of a degenerate elliptic boundary-value problem
zbMATH Open document ID: 1113.35067
DOI: 10.4171/RLM/471
author: Gilles Evéquoz
author: C. A. Stuart
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2007-07-19T00:00:00Z
review text: The authors study the nonlinear degenerate elliptic boundary value problem \[ - \nabla [A(x) \nabla u(x)]= \lambda f( u(x) ),\quad x \in \Omega \] \[ u(x)=0,\qquad x\in \partial \Omega \] \(\Omega\) being a bounded open subset of \(\mathbb R^N\), \(N \geq 3,\) having Lipschitz boundary. The ellipticity is degenerate in the sense that \(0\in\Omega \) and \(A(x)>0 \) for all \(x\) not equal to zero, but \(\lim_{x \to 0}\frac{A(x)}{x^2}=1\). In the note it is proved that there is a vertical bifurcation in all points \(\lambda\in [N^2/4,\infty)\) that also occurs at all the eigenvalues of the linearized problem that are below \(N^2/4\). We point out that the approach is based on a recent study by the authors concerning problems that are differentiable in the sense of Hadamard, but not in the sense of Fréchet.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 58E07
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 5173046
zbMATH Keywords: Degenerate elliptic equation
zbMATH Keywords: Bifurcation
zbMATH Keywords: Hadamard differentiable
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5173046