Precise spectral asymptotics for nonlinear Sturm-Liouville problems
type: Item
instance of: scholarly article
title: Precise spectral asymptotics for nonlinear Sturm-Liouville problems
zbMATH Open document ID: 1021.34068
author: Tetsutaro Shibata
published in: Journal of Differential Equations
publication date: 2003-10-12T00:00:00Z
review text: Given a constant \(p>1\), it is known from the work of \textit{H. Berestycki} [J. Funct. Anal. 40, 1-29 (1981; Zbl 0452.35038)] and others that, for each \(\alpha >0\), there is a unique solution \((\lambda ,u)=(\lambda (\alpha),u_{\alpha})\in {\mathbb{R}}_+ \times C^2 (\bar{I})\) of \(-u'' +u^p =\lambda u\) on \(I:=(0,1)\) satisfying \(u_{\alpha}>0\) on \(I\), \(u_{\alpha}(0)=u_{\alpha}(1)=0\) and \(\|u_{\alpha}\|_2 =\alpha\). This paper establishes the asymptotic expansions of \(\lambda (\alpha)\), \(u' _{\alpha}(0)^2\) and \(\|u_{\alpha}\|_{\infty}^{p-1}\) as \(\alpha \to \infty\). As a corollary, some analogous results are proved for \(-u''+|u|^{p-1}u=\lambda u\) on \(I\), \(u(0)=u(1)=0\), without the sign restriction.
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34L30
zbMATH DE Number: 1750349
zbMATH Keywords: nonlinear Sturm-Liouville problems
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: boundary layers
reviewed by: Alan L. Andrew
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4061
OpenAlex ID: W2002888170
cites work: Le nombre de solutions de certains problèmes semi-linéaires elliptiques
cites work: Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems of unbounded domains
cites work: On nonlinear eigenvalue problems
cites work: Remarks on bifurcation for elliptic operators with odd nonlinearity
cites work: On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term
cites work: Constrained critical points and eigenvalue approximation for semilinear elliptic operators
cites work: On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems
cites work: Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems
cites work: Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems
cites work: Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems
cites work: Uniqueness of global positive solution branches of nonlinear elliptic problems
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: Asymptotic behavior of the variational eigenvalues for semilinear sturm-liouville problems
DOI: 10.1006/JDEQ.2001.4061
description: scientific article; zbMATH DE number 1750349