Characterisation of bifurcation from the continuous spectrum by nodal properties
type: Item
instance of: scholarly article
title: Characterisation of bifurcation from the continuous spectrum by nodal properties
zbMATH Open document ID: 0547.34018
DOI: 10.1016/0022-0396(84)90158-X
author: Christopher K. R. T. Jones
author: Tassilo Küpper
published in: Journal of Differential Equations
publication date: 1984-01-01T00:00:00Z
review text: By variational methods it was known that the nonlinear eigenvalue problem \(-u''+w(x)| u|^{\sigma}u=\lambda u, u(0)=0\), \(u\in L^ 2(0,\infty)\) has infinitely many solutions for each \(\lambda >0\), if w is positive, \(\sigma >0\) and \(\int^{\infty}_{0}w^{- 2/\sigma}(x)dx<\infty\). Here it is shown for exponentially growing w that these solutions can be characterised by the number of zeros. For \(\lambda \to 0\) all solutions bifurcate from the lowest point of the continuous spectrum and this characterisation gives that there are infinitely many branches characterised by nodal properties. The proof is based on a dynamical systems approach which was recently extended to semilinear equations in \({\mathbb{R}}^ n\).
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 3871795
zbMATH Keywords: variational methods
zbMATH Keywords: continuous spectrum
zbMATH Keywords: characterisation
zbMATH Keywords: nodal properties
MaRDI profile type: Publication
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full work available at URL: 0022-0396(84)90158-x
OpenAlex ID: W2019212573
description: scientific article; zbMATH DE number 3871795