Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities
type: Item
instance of: scholarly article
title: Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities
zbMATH Open document ID: 0837.34033
author: Philip Korman
author: Tiancheng Ouyang
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-05-28T00:00:00Z
review text: The paper deals with multiplicity results for the nonlinear Dirichlet boundary value problem \((*)\) \(u'' + \lambda u (u - a(x)) (b(x) - u) = 0\) \(u (-1) = 0 = u(1)\). Conditions on the functions \(a,b\) are given which guarantee that only the two following possibilities can occur: (i) BVP \((*)\) has no solution for any \(\lambda > 0\). (ii) There exists \(\lambda_0\) such that \((*)\) has either zero, one, or two solutions depending on whether \(\lambda < \lambda_0\), \(\lambda = \lambda_0\), or \(\lambda > \lambda_0\), respectively. The principal method used in the proof of this statement is the Crandall-Rabinowitz bifurcation theorem.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 811238
zbMATH Keywords: cubic nonlinearity
zbMATH Keywords: multiplicity results
zbMATH Keywords: nonlinear Dirichlet boundary value problem
zbMATH Keywords: Crandall-Rabinowitz bifurcation theorem
reviewed by: Ondřej Došlý
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1302
OpenAlex ID: W1965781610
DOI: 10.1006/JMAA.1995.1302
description: scientific article; zbMATH DE number 811238