A complete classification of bifurcation diagrams of classes of a multiparameter Dirichlet problem with concave-convex nonlinearities
type: Item
instance of: scholarly article
title: A complete classification of bifurcation diagrams of classes of a multiparameter Dirichlet problem with concave-convex nonlinearities
zbMATH Open document ID: 1160.34017
author: Kuo-Chih Hung
author: Shin-Hwa Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-11-06T00:00:00Z
review text: The authors study the bifurcation diagrams for positive solutions of the multi-parameter Dirichlet problem \[ u^{''}(x)+f_{\lambda,\mu} (u(x))=0, \quad-1<t<1, \] \[ u(-1)=u(1)=0, \] where \(f_{\lambda,\mu}(u)=g(u,\lambda)+h(u,\mu)\), \(\lambda>\lambda_0\), \(\mu>\mu_0\) are two bifurcation parameters, \(\lambda_0\) and \(\mu_0\) are two given real numbers. Under some reasonable conditions on \(g\) and \(f\), they give a classification of totally \textbf{eight} qualitatively different bifurcation diagrams, and prove the exact multiplicity of positive solutions. In addition, they give interesting examples which show complete evolution of bifurcation diagrams as \(\mu\) (resp. \(\lambda\)) varies. For earlier work, see \textit{A. Ambrosetti} et al. [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)].
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5362140
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: positive solution
zbMATH Keywords: exact multiplicity
zbMATH Keywords: Dirichlet problem
reviewed by: Ruyun Ma
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.08.020
OpenAlex ID: W2065255289
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cites work: A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities.
DOI: 10.1016/J.JMAA.2008.08.020
description: scientific article; zbMATH DE number 5362140