Eigenvalue asymptotics and a nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Eigenvalue asymptotics and a nonlinear Schrödinger equation
zbMATH Open document ID: 0786.35097
DOI: 10.1007/BF02761304
published in: Israel Journal of Mathematics
publication date: 1993-09-07T00:00:00Z
review text: Let \(S\) be the unit sphere in \(L^ 2(\mathbb{R}^ N)\), \(N \geq 3\) and \(r>0\). The authors show the existence of infinitely many eigenvalues \(\lambda_ n\) of the problem \[ -\Delta u+qu+f(x,u)=\lambda u \text{ in }\mathbb{R}^ N,\qquad u\in W^{1,2}(\mathbb{R}^ N)\cap rS. \] The asymptotic behavior of \(\lambda_ n\) as \(n \to \infty\) is also studied. The function \(q\) is assumed to be bounded from below, \(q(x) \to \infty\) as \(| x | \to \infty\), the perturbation term \(f\) is assumed to be a continuous function, \(f(x,-s)=-f(x,s)\), \(f(x,s)s \geq 0\) and \(| f(x,s)| \leq a | s |^{p-1}+b(x)\), where \(b \in L^ 2(\mathbb{R}^ N)\) and \(p<2+ 4/N\). The proofs use the Lyusternik-Schnirelman theory and they rely on the compactness of the embedding of the weighted Sobolev space \(W_ q^{1,2} (\mathbb{R}^ N)=\{u;\int_{\mathbb{R}^ N}(| \nabla u |^ 2+qu^ 2)dx<\infty\}\) in \(L^ p(\mathbb{R}^ N)\).
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 410034
zbMATH Keywords: existence of infinitely many eigenvalues
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Lyusternik-Schnirelman theory
zbMATH Keywords: embedding of the weighted Sobolev space
author: Raffaele Chiappinelli
author: D. E. Edmunds
reviewed by: Pavol Quittner
MaRDI profile type: Publication
cites work: Q4146134
cites work: Q5612123
cites work: Q3729473
cites work: Q3976140
cites work: Q3764747
cites work: Q3847203
cites work: Asymptotic distribution of Liusternik-Schnirelman eigenvalues for elliptic nonlinear operators
cites work: Some aspects of nonlinear eigenvalue problems
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description: scientific article; zbMATH DE number 410034