Exact multiplicity for degenerate two-point boundary value problems with \(p\)-convex nonlinearity
type: Item
instance of: scholarly article
title: Exact multiplicity for degenerate two-point boundary value problems with \(p\)-convex nonlinearity
zbMATH Open document ID: 1028.34024
DOI: 10.1016/S0362-546X(02)00268-7
author: János Karátson
author: Péter L. Simon
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The authors investigate the exact number of positive solutions to degenerate two-point boundary value problems \[ (|u'|^{p-2}u')'+f(u)=0,\;-R<x<R,\;u(-R)=u(R)=0, \] under the assumption that \(f\) is p-convex, which is a generalization of convexity. They obtain the following result: Suppose \(f\) is \(C^2,\) strictly p-convex and \(f(u)/u^{p-1}\) is infinite when u is infinite. If \(f(0)\leq 0\), then the problem has a unique solution. If \(f(0)>0\), then there are \(0,1\) or \(2\) solutions provided that the number of solutions is finite. The proofs are based on the shooting method for the characterization of the shape of the time-map.
reviewed by: Abdelkader Boucherif
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1880112
zbMATH Keywords: degenerate two-point boundary value problems
zbMATH Keywords: time-map
zbMATH Keywords: p-convex nonlinearity
zbMATH Keywords: bifurcation diagram of positive solutions
zbMATH Keywords: dead core
MaRDI profile type: Publication
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full work available at URL: s0362-546x(02)00268-7
OpenAlex ID: W2085371211
description: scientific article; zbMATH DE number 1880112