Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems
type: Item
instance of: scholarly article
title: Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems
zbMATH Open document ID: 0712.34035
DOI: 10.1007/BF00375700
published in: Archive for Rational Mechanics and Analysis
publication date: 1990-01-01T00:00:00Z
review text: The author considers a class of nonlinear Sturm-Liouville problems \[ (0.1)\quad -u''=\lambda r(x)u-F(x,u)u,\quad a<x<+\infty,\quad u(a)\cos \theta -u'(a)\sin \theta =0,\quad u\in L^ 2. \] It is easy to see that (0.1) admits zero as a trivial solution \(\forall \lambda\) and that the linearization around zero has purely continuous spectrum \([0,+\infty)\). For a special type of nonlinearity \(F(x,u)=\psi (w(x)| u|^{\sigma})\) with \(\psi (t)\geq pt^ q\), \(\psi '>0\), \(w'(0)\geq 0\), \(w'/w\) nondecreasing, the author proves the uniqueness of solutions having a prescribed number of nodes. From this result it is possible to deduce that there are infinitely many continuous curves of solutions, bifurcating from the origin parametrized by \(\lambda\) and characterized by their nodal properties. Further, for a more general class of nonlinearities the author proves a bifurcation theorem analogous to the preceding one, where the continuous curves are replaced by unbounded connected components.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 4171390
zbMATH Keywords: bifurcation
zbMATH Keywords: nonlinear Sturm-Liouville problems
zbMATH Keywords: number of nodes
author: Chao-Nien Chen
reviewed by: Pierluigi Zezza
MaRDI profile type: Publication
cites work: Q3684533
cites work: On solitary water-waves of finite amplitude
cites work: Does bifurcation from the essential spectrum occur?
cites work: Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems of unbounded domains
cites work: Bifurcation when the linearized problem has no eigenvalues
cites work: Q5848416
cites work: Q5343623
cites work: Q5820676
cites work: Q5606061
cites work: The \(L^2\) solutions of linear differential equations of second order
cites work: Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems
cites work: Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems
cites work: Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems
cites work: Q5631509
cites work: Characterisation of bifurcation from the continuous spectrum by nodal properties
cites work: Heavy rotating string—a nonlinear eigenvalue problem
cites work: On minimal nonlinearities which permit bifurcation from the continuous spectrum
cites work: A note on a nonlinear eigenvalue problem for a class of differential equations
cites work: Nonlinear sturm‐lionville problems for second order ordinary differential equations
cites work: Some global results for nonlinear eigenvalue problems
cites work: Some aspects of nonlinear eigenvalue problems
cites work: Q3308426
cites work: Bifurcation for Dirichlet Problems Without Eigenvalues
cites work: Bifurcation for Neumann problems without eigenvalues
cites work: Q4090688
full work available at URL: https://doi.org/10.1007/bf00375700
OpenAlex ID: W2076373950
description: scientific article; zbMATH DE number 4171390