Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states
type: Item
instance of: scholarly article
title: Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states
zbMATH Open document ID: 0958.34017
author: H. Jeanjean
author: C. A. Stuart
published in: Advances in Differential Equations
publication date: 2000-08-21T00:00:00Z
review text: The authors consider the nonlinear eigenvalue problem \[ u''(x)+\lambda u(x)+f(x,u(x))=0, x \in {\mathbb{R}}; \;\lim_{\mid x \mid \to \infty}{u(x)}=0, u(x) \geq 0; x \in {\mathbb{R}}; \;u \not \equiv 0. \] They prove the existence of a curve of positive solutions which bifurcates from the lowest eigenvalue of the linearized problem. This curve can be parametrized globally by \(\lambda\) and can be extended to infinity. They establish that all solutions to the equation are even and monotone and under appropriate conditions, all of them belong to the curve of bifurcation. The obtained results depend on the conditions of symmetry and monotonicity imposed to \(f\).
reviewed by: Alberto Cabada
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 58E07
Mathematics Subject Classification ID: 78A50
Mathematics Subject Classification ID: 78A60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J15
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 1493611
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: bifurcation theory
zbMATH Keywords: positive solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1493611