Multiple solutions for a class of nonlinear Sturm-Liouville problems on the half line
type: Item
instance of: scholarly article
title: Multiple solutions for a class of nonlinear Sturm-Liouville problems on the half line
zbMATH Open document ID: 0703.34032
DOI: 10.1016/0022-0396(90)90117-8
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: The author considers a class of nonlinear Sturm-Liouville problems. He gives an extensive list of references containing fourty references together with the complete proofs of the following: 7 theorems, 17 remarks, 17 lemmas, 5 propositions and two corollaries which are pieced together to construct essentially the author's doctoral thesis from the University of Wisconsin Madison (USA) where the reviewer worked as a visiting professor for fourteen months. In the first section the author obtains existence and uniqueness for positive and negative solutions for the problems \[ (1)\quad -u^ n=\lambda r(x)u-F(x,u)u,\quad 0<x<+\infty,\quad u(a)\cos \theta -u'(a) \sin \theta =0,\quad u\in L^ 2[a,\infty), \] where r and F are nonnegative continuous functions, \(a\geq 0\) and \(0\leq \theta \leq \pi /2.\) In section 2 he obtains the existence of solutions with prescribed number of nodes when the nonlinearitiy is odd near zero. In section 3 the author considers analogous results for radical solutions in higher dimensional class, while in section 4 he considers the partial differential equation \[ (2)\quad \Delta \hat u=\lambda \hat r(x)\hat u(x)-\hat F(x,\hat u(x))\hat u(x),\quad x\in {\mathbb{R}}, \] where \(\hat r:\) \({\mathbb{R}}^ N\to (0,\infty)\) and \(\hat F:\) \({\mathbb{R}}^ N\times {\mathbb{R}}\to (0,\infty)\) are radially symmetric. The author seeks the solution of (2) in the form \(\hat u\in L^ 2({\mathbb{R}}^ N)\cap C^ 2({\mathbb{R}}^ N)\).
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4152874
zbMATH Keywords: nonlinear Sturm-Liouville problems
author: Chao-Nien Chen
reviewed by: F. M. Ragab
MaRDI profile type: Publication
full work available at URL: 0022-0396(90)90117-8
OpenAlex ID: W2086245106
cites work: Q3684533
cites work: Does bifurcation from the essential spectrum occur?
cites work: Nonlinear scalar field equations. I: Existence of a ground state
cites work: Q5574850
cites work: Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems of unbounded domains
cites work: Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems
cites work: Multiple solutions for a class of nonlinear Sturm-Liouville problems when nonlinearities are not odd
cites work: Bifurcation when the linearized problem has no eigenvalues
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q3999241
cites work: Q5606061
cites work: Multiple solutions of semilinear elliptic problems in a ball
cites work: Q4040640
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: On the Infinitely Many Solutions of a Semilinear Elliptic Equation
cites work: The lowest point of the continuous spectrum as a bifurcation point
cites work: On minimal nonlinearities which permit bifurcation from the continuous spectrum
cites work: Necessary and sufficient conditions for bifurcation from the continuous spectrum
cites work: Characteristic values associated with a class of nonlinear second-order differential equations
cites work: Q5533735
cites work: A note on a nonlinear eigenvalue problem for a class of differential equations
cites work: Some global results for nonlinear eigenvalue problems
cites work: Nonlinear sturm‐lionville problems for second order ordinary differential equations
cites work: Some aspects of nonlinear eigenvalue problems
cites work: Q5526189
cites work: Boundary value problems for a class of nonlinear differential equations
cites work: Q4347868
cites work: Bifurcation for Dirichlet Problems Without Eigenvalues
cites work: Bifurcation for Neumann problems without eigenvalues
cites work: Q4090688
cites work: Global bifurcation for Neumann problems without eigenvalues
cites work: Superlinear Sturm-Liouville problems
description: scientific article; zbMATH DE number 4152874