A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities.
type: Item
instance of: scholarly article
title: A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities.
zbMATH Open document ID: 1054.34040
author: Shin-Hwa Wang
author: Tzung-Shin Yeh
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: The authors study the bifurcation diagrams of positive solutions of the two-point boundary value problem \(u''(x)+\lambda{g(u(x))}+h(u(x))=0\), \(-1<x<1\), \(u(-1)=u(1)=0\), where \(g\) and \(h\) are continuous on \([0,+\infty)\) and twice continuously differentiable on \((0,+\infty)\). Moreover, \(g\) is concave and \(h\) is convex. \(\lambda\) is a positive bifurcation parameter. A complete classification of the bifurcation diagrams is given. The authors prove an exact multiplicity result for positive solutions. The paper has links to a paper by \textit{A. Ambrosetti}, \textit{H. Brézis} and \textit{G. Cerami} [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)].
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 2056500
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: positive solution
zbMATH Keywords: exact multiplicity
zbMATH Keywords: solution curve
zbMATH Keywords: concave-convex nonlinearity
zbMATH Keywords: time map
reviewed by: Abdelkader Boucherif
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.021
OpenAlex ID: W2111152667
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DOI: 10.1016/J.JMAA.2003.10.021
description: scientific article; zbMATH DE number 2056500