The structure of the solution set of a generalized Ambrosetti--Brezis--Cerami problem in one space variable
type: Item
instance of: scholarly article
title: The structure of the solution set of a generalized Ambrosetti--Brezis--Cerami problem in one space variable
zbMATH Open document ID: 1102.34014
author: Wen-Yin Hsia
author: Shin-Hwa Wang
author: Tzung-Shin Yeh
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: This paper deals with the structure of the set of positive solutions of the nonlinear two-point boundary value problem \[ u''(x) + f_\lambda (u(x)) = 0\,, \quad -1 < x < 1,\quad u(-1) = u(1) = 0, \] where \(\lambda > 0\) is a bifurcation parameter and \[ f_\lambda(u) = \lambda \sum_{i=1}^m a_i u^{q_i} + \sum_{j=1}^n b_j u^{p_j} \] satisfies certain conditions which do not involve sign conditions on the coefficients \(a_i, b_j\). Using classical ODE methods based on the study of an appropriate time map, the authors give a precise description of the bifurcation diagram. In particular, they prove that there exists \(\lambda^* > 0\) such that the problem has exactly two positive solutions for \(0 < \lambda < \lambda^*\), exactly one positive solution for \(\lambda = \lambda^*\), and no positive solution for \(\lambda > \lambda^*\).
reviewed by: Hans-Peter Heinz
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5016112
zbMATH Keywords: solution set
zbMATH Keywords: exact multiplicity
zbMATH Keywords: positive solution
zbMATH Keywords: bifurcation
zbMATH Keywords: concave-convex nonlinearity
zbMATH Keywords: time map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.03.077
OpenAlex ID: W2158572207
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DOI: 10.1016/J.JMAA.2005.03.077
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description: scientific article; zbMATH DE number 5016112