On the number of solutions of nonlinear Schrödinger equations and on unique continuation
type: Item
instance of: scholarly article
title: On the number of solutions of nonlinear Schrödinger equations and on unique continuation
zbMATH Open document ID: 0840.35105
author: Hans-Peter Heinz
published in: Journal of Differential Equations
publication date: 1996-07-04T00:00:00Z
review text: Consider the following equations: \[ (- \Delta+ q(x)) u= \lambda u\pm f(x, u), \] where \(q\in L^\infty(\mathbb{R}^n)\) is real-valued and where \(f: \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}\) satisfies ``quasi-polynomials'' conditions. The author studies critical values of \(\varepsilon\) for the stability of radial solutions. Since \(q\in L^\infty\), the operator \(- \Delta+ q(x)\) has a unique selfadjoint extension \(H\) and the equation above is written as \[ (H- \lambda) u= \pm f(x, u). \] The author derives that there are infinitely many non-zero weak solutions, if \(\lambda\) is in the ``spectral gap'' and \(I(f):= \{x\in \mathbb{R}^n\mid f(x, s)\neq 0 \forall s\}\) has positive Lebesgue measure, and obtains bounds of these weak solutions.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B60
zbMATH DE Number: 729101
zbMATH Keywords: stability of radial solutions
zbMATH Keywords: infinitely many non-zero weak solutions
zbMATH Keywords: bounds
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1032
OpenAlex ID: W1995434318
DOI: 10.1006/JDEQ.1995.1032
description: scientific article; zbMATH DE number 729101