Multiplicity and concentration for the nonlinear Schrödinger equation with critical frequency
type: Item
instance of: scholarly article
title: Multiplicity and concentration for the nonlinear Schrödinger equation with critical frequency
zbMATH Open document ID: 1124.35083
author: P. Felmer
author: Juan Mayorga-Zambrano
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-01-09T00:00:00Z
review text: The authors consider the nonlinear Schrödinger equation \(\epsilon{^{2}}\Delta v-V(x)v+| v| {^{p-1}}v=0\) in \({\mathbb{R}}{^{n}}\) with \(\epsilon>0,\) \(n\geq 3,\) and \(p\in(1,(n+2)/(n-2)),\) where \(V\) is a nonnegative, continuous potential that becomes infinite as \(| x| \to \infty.\) By applying the Ljusternik-Schnirelman theory, it is shown that there are infinitely many solutions to this equation and such solutions decay exponentially as \(| x| \to \infty.\) It is also shown that, as \(\epsilon\to 0,\) such solutions satisfy \(\Delta u+| u| {^{p-1}}u=0\) in a bounded, connected set \(\Omega\) with \(u=0\) on the boundary of \(\Omega.\) A similar problem has been analyzed by \textit{Y. Ding} and \textit{A. Szulkin} [in: T. Cazenave et al. (eds.), Contributions to nonlinear analysis, Birkhäuser, Basel, pp. 221--231 (2006; Zbl 1131.35332)].
reviewed by: Tuncay Aktosun
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5083403
zbMATH Keywords: critical point theory
zbMATH Keywords: Ljusternik-Schnirelman theory
MaRDI profile type: Publication
full work available at URL: j.na.2005.11.017
OpenAlex ID: W2140805941
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DOI: 10.1016/J.NA.2005.11.017
description: scientific article; zbMATH DE number 5083403