A local bifurcation theorem for degenerate elliptic equations with radial symmetry
type: Item
instance of: scholarly article
title: A local bifurcation theorem for degenerate elliptic equations with radial symmetry
zbMATH Open document ID: 1017.34040
published in: Journal of Differential Equations
publication date: 2002-07-02T00:00:00Z
review text: The following bifurcation problem is considered \[ -\text{ div }(|\text{ grad }u|^{p-2}\text{ grad }u)=\lambda |u|^{p-2}u+g(\lambda,x,u)\quad\text{ in } B,\quad u=0\quad\text{ on }B, \] where \(p>1\), \(B\subset \mathbb{R}^N\) is a ball with radius \(1\) and centered at the origin, \(g\in C^1\), \(g(\lambda,x,0)\equiv 0\) and \(\partial g/\partial u=o(|u^{p-2}|)\) as \(u\to 0\) and \(\lambda\) is the spectral parameter. Radial solutions are considered only reducing the problem to an ODE. The couple \((\lambda,u)\) is understood as a solution. With \(\{\lambda_{n,p}\}\) being radial eigenvalues of the p-Laplacian on \(B\), main results of the paper can be stated as that the only nontrivial solutions close to the trivial ones \((\lambda_{n,p},0)\) (for \(n,p\) fixed) lie on a continuous curve, thus extending the Crandall-Rabinowitz theorem to degenerated elliptic problems.
reviewed by: Petr Girg
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 1744184
zbMATH Keywords: degenerate quasilinear elliptic equations
zbMATH Keywords: p-Laplacian
zbMATH Keywords: local bifurcation theory
zbMATH Keywords: Crandall-Rabinowitz theorem
author: Jorge García-Melian
author: José C. Sabina de Lis
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4031
OpenAlex ID: W2039908209
cites work: A note on radiality of solutions of P-laplacian
cites work: Q5533446
cites work: Bifurcation from simple eigenvalues
cites work: Global bifurcation from the eigenvalues of the \(p\)-Laplacian
cites work: On the global bifurcation for a class of degenerate equations
cites work: Bifurcation of radially symmetric solutions of degenerate quasilinear elliptic equations
cites work: Some bifurcation results for a class of \(p\)-Laplacian like operators
cites work: Uniqueness to quasilinear problems for the \(p\)-Laplacian in radially symmetric domains
cites work: Stationary profiles of degenerate problems when a parameter is large
cites work: Symmetry and related properties via the maximum principle
cites work: Q4040640
cites work: Bifurcation Phenomena Associated to the p-Laplace Operator
cites work: Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: Global bifurcation for quasilinear elliptic equations
cites work: Some global results for nonlinear eigenvalue problems
cites work: On The Dirichletproblem for Quasilinear Equations
cites work: A strong maximum principle for some quasilinear elliptic equations
DOI: 10.1006/JDEQ.2001.4031
description: scientific article; zbMATH DE number 1744184