Localizing Estimates of the Support of Solutions of some Nonlinear Schr\"{o}dinger Equations - The Stationary Case

TL;DR

This paper investigates how solutions to certain nonlinear Schrödinger equations can have compact support, revealing conditions under which solutions localize spatially, contrasting with linear cases where solutions cannot vanish on positive measure sets.

Contribution

The authors demonstrate that singular nonlinear terms can lead to solutions with compact support, extending previous results with new energy methods and nonlinear differential inequalities.

Findings

Solutions can have compact support under certain nonlinear perturbations

Energy methods relate support compactness to local vanishing of an energy function

Results extend previous work from 2006

Abstract

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schr\"{o}dinger equations mainly the compactness of the support and its spatial localization. This question is very related with pure essence of the derivation of the Schr\"{o}dinger equation since it is well-known that if the linear Schr\"{o}dinger equation is perturbated with "regular" potentials then the corresponding solution never vanishes on a positive measured subset of the domain, which corresponds with the impossibility of localize the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution becomes a compact set and so any estimate on its spatial localization implies a very rich information on places which can not be. Our results are obtained by the application of some energy methods which connect the compactness of the support with the local vanishing of a suitable "energy function" which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.