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Localizing estimates of the support of solutions of some nonlinear Schrodinger equations - The stationary case

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Díaz Díaz, Jesús Ildefonso and Begout, Pascal (2012) Localizing estimates of the support of solutions of some nonlinear Schrodinger equations - The stationary case. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 29 (1). pp. 35-38. ISSN 0294-1449

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Official URL: http://www.sciencedirect.com/science/article/pii/S0294144911000837


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Abstract

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrodinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrodinger equation, since it is well-known a solution of a linear Schrodinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain 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.


Item Type:Article
Uncontrolled Keywords:singular complex potentials; operators; Nonlinear Schrodinger equation; Compact support; Energy method
Subjects:Sciences > Mathematics > Differential equations
ID Code:15006
Deposited On:25 Apr 2012 08:44
Last Modified:12 Dec 2018 15:07

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