Perturbed \(p\)-Laplacian in \(\mathbb{R}^ N\): Bifurcation from the principal eigenvalue
type: Item
instance of: scholarly article
title: Perturbed \(p\)-Laplacian in \(\mathbb{R}^ N\): Bifurcation from the principal eigenvalue
zbMATH Open document ID: 0874.35034
author: Pavel Drábek
author: Yin Xi Huang
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-10-22T00:00:00Z
review text: The authors consider the perturbed bifurcation problem in \(\mathbb{R}^N\), \[ - div (a(x,u){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u + f(\lambda, x, u), \eqno (1) \] and the associated problem with homogeneous principal part, \[ - div (a_{0} (x){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u + f(\lambda, x, u), \eqno (2) \] where \(1 < p < N\) ; \(a\) and \(a_{0}\) are both positive in \(\mathbb{R}^N\) and may be singular or degenerate at infinity, no growth restriction on \(a (x, \cdot)\) is postulated, and both \(f\) and \(g\) may change sign. The authors prove that the principal eigenvalue \(\lambda_{1}\) of the homogeneous eigenvalue problem \(- div (a_{0} (x){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u\) is a bifurcation point for (1) and (2). \noindent This paper extends earlier results from the same authors.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 1005654
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: perturbed bifurcation problem
reviewed by: Jacqueline Fleckinger
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0455
OpenAlex ID: W1979953721
DOI: 10.1006/JMAA.1996.0455
description: scientific article; zbMATH DE number 1005654