Variational method for precise asymptotic formulas for nonlinear eigenvalue problems
type: Item
instance of: scholarly article
title: Variational method for precise asymptotic formulas for nonlinear eigenvalue problems
zbMATH Open document ID: 1069.34026
DOI: 10.1007/BF03322876
author: Tetsutaro Shibata
published in: Results in Mathematics
publication date: 2005-02-11T00:00:00Z
review text: The following nonlinear eigenvalue problem is considered \[ \begin{alignedat}{2} -&u''(t)+ f(u(t))=\lambda u(t)^p, && \quad t\in I:=(-T,T),\\ & u(t)>0, && \quad t\in I,\\ & u(\pm T)=0, \end{alignedat} \] where \(\lambda\in\mathbb{R}\) is a parameter and \(p>1\). The main hypotheses on the nonlinearity \(f\) are the following: (i). \(f\in C^1(\mathbb{R}),\;f(-u)=-f(u)\). (ii). \(| f(u)| /u\rightarrow 0\) as \(u\rightarrow\infty\). The author investigates the relationship between \(\lambda\), the \(L^\infty\) norm of the solutions, their slope on the boundary and their \(L^{p+1}\) norm, especially when the later is large enough. To achieve this goal an associated variational problem with the equation is considered on \[ M_\alpha=\left\{ v\in H^1_0(I):\;| | v| | _{L^{p+1}}=\alpha\right\}, \] where \(\alpha >0\) is a parameter. After the existence of a solution pair \((\lambda(\alpha), u_\alpha)\in \mathbb{R}\times M_\alpha\) for this problem has been established, precise asymptotic formulas are obtained for \(| | u_\alpha| | _\infty,\;u'_\alpha(\pm T)^2\) and \(\lambda(\alpha)\) as \(\alpha\rightarrow\infty\).
reviewed by: Nikos Yannakakis
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2133442
zbMATH Keywords: precise aymptotic formula
zbMATH Keywords: variational method
zbMATH Keywords: nolinear eigenvalue
zbMATH Keywords: Lagrange multiplier
MaRDI profile type: Publication
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
description: scientific article; zbMATH DE number 2133442