Semi-classical limit for radial nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Semi-classical limit for radial nonlinear Schrödinger equation
zbMATH Open document ID: 1075.35010
author: Rodrigo Castro
author: P. Felmer
published in: Communications in Mathematical Physics
publication date: 2005-06-08T00:00:00Z
review text: The authors study the Schrödinger equation \(\varepsilon^{2}\Delta - V(x)u + | u |^{p-1}u=0\) in \(\mathbb{R}^{N}\). The potential \(V\) is radially symmetric, of class \(C^{1}\), uniformly continuous and such that \(\liminf_{r\rightarrow\infty}V(r) > 0\). It is also assumed that the effective potential \(U(r)=r^{\alpha(p-1)}V(r)\), \(\alpha={2(N-1)\over{p+3}}\) is either an increasing positive function with derivative uniformly bounded away from \(0\) for \(r\geq1\) or is such that there exist \(d>0\), \(\eta>0\) with \(U(r)(r-d)>0\) for \(r>0\), \(r\not=d\) and \(U'(r)\geq \eta\) for \(r\geq d\). One proves that if \(p\in (\min(5,{{N+2}\over{N-2}}))\) then one can find a sequence \(\varepsilon_{n}\rightarrow 0\) and corresponding radial solutions \(u_{n}\) of the equation such that \(| u_{n}(r) | \leq Cr^{-\alpha}\) for all positive integers \(n\) and \(r>0\) and for some positive constant \(C\). The zeros of \(u_{n}\) become dense in a neighborhood of the origin when \(n\rightarrow \infty\). Also the approximate envelope functions of \(u_{n}\) converge to some envelope function which characterize the asymptotic behavior of the solutions. The envelope function is the unique solution of a first order differential equation subject to some additional condition.
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 2174487
zbMATH Keywords: radial sign-changing solutions
zbMATH Keywords: envelope functions
MaRDI profile type: Publication
full work available at URL: s00220-005-1320-y
OpenAlex ID: W2063250217
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DOI: 10.1007/S00220-005-1320-Y
description: scientific article; zbMATH DE number 2174487