A pointwise lower bound for positive solutions of a Schrödinger equation in \(\mathbb{R}^ N\)
type: Item
instance of: scholarly article
title: A pointwise lower bound for positive solutions of a Schrödinger equation in \(\mathbb{R}^ N\)
zbMATH Open document ID: 0874.35030
author: Bénédicte Alziary
published in: Journal of Differential Equations
publication date: 1997-04-27T00:00:00Z
review text: It is first shown that \((-\Delta+ q(x))u= f\geq 0\), \(f\neq 0\), implies \(u\geq c \varphi_1\), where \(\varphi_1\) is the first eigenfunction and \(c>0\) is a constant. Here, the potential has to grow at infinity, typically like \(|x|^{z +\varepsilon}\) with \(\varepsilon>0\), and it has to be a perturbation of a radially symmetric function. For the proof, weak and strong maximum principles in exterior domains are used. The result is sharp in the sense that it does not extend to the harmonic oscillator, i.e. \(\varepsilon=0\), as is demonstrated, too. Finally, applications to semilinear Schrödinger equations are given.
reviewed by: Reinhard Racke
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1000154
zbMATH Keywords: comparison principle
zbMATH Keywords: maximum principles in exterior domains
zbMATH Keywords: harmonic oscillator
zbMATH Keywords: semilinear Schrödinger equations
MaRDI profile type: Publication
full work available at URL: jdeq.1996.3205
OpenAlex ID: W2051143119
DOI: 10.1006/JDEQ.1996.3205
description: scientific article; zbMATH DE number 1000154