Spectral properties of \(p\)-Laplacian problems with Neumann and mixed-type multi-point boundary conditions
type: Item
instance of: scholarly article
title: Spectral properties of \(p\)-Laplacian problems with Neumann and mixed-type multi-point boundary conditions
zbMATH Open document ID: 1215.34106
author: Bryan P. Rynne
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-01-31T00:00:00Z
review text: The author first investigates the spectral problem of the \(p\)-Laplacian eigenvalue problem \[ -\phi_p(u')'(x)=\lambda\phi_p(u)(x),\quad0<x<1 \] together with Dirichlet-type, Neumann-type or mixed boundary conditions. Here, \(\phi_p(s)=| s|^{p-2}s\) is the \(p\)-Laplacian operator (\(p>1\)) and \(\lambda\) corresponds to an eigenvalue of the problem. Then the degree of an associated operator is defined and computed. Several applications are provided in this paper. First, a global bifurcation theory is developed for the nonlinear problem \[ -\phi_p(u')'(x)=\lambda f(u)(x),\quad 0<x<1, \] where \(f: \mathbb{R}\rightarrow\mathbb{R}\) is continuous and satisfies the sign condition \(sf(s)>0\) for \(s\in\mathbb{R}^*\) both with the growth condition \[ 0<\lim_{s\to0}\frac{f(s)}{\phi_p(s)}<\infty. \] The author proves Rabinowitz-type global bifurcation results. Then under a non-resonance condition, the existence of a solution is obtained for the nonlinear equation \[ -\phi_p(u')'=f(u)+h \] and also the existence of nodal solutions in case \(h=0\) is obtained. The paper ends with some properties of the spectrum of the operator \(-\phi_p(u')'\).
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5843311
zbMATH Keywords: \(p\)-Laplacian operator
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: bifurcation problem
zbMATH Keywords: nonresonance conditions
zbMATH Keywords: mixed boundary conditions
reviewed by: Smail Djebali
MaRDI profile type: Publication
full work available at URL: j.na.2010.10.020
OpenAlex ID: W2030550554
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DOI: 10.1016/J.NA.2010.10.020
description: scientific article; zbMATH DE number 5843311