On a nonlinear Schrödinger equation with a localizing effect

Bégout, Pascal; Díaz, Jesús Ildefonso

doi:10.1016/j.crma.2006.01.027

Partial Differential Equations

On a nonlinear Schrödinger equation with a localizing effect
[Sur une équation de Schrödinger non-linéaire avec un effet localisant]

Présenté par : Haïm Brezis

Bégout, Pascal ¹ ; Díaz, Jesús Ildefonso ²

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
² Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias, 3, 28040 Madrid, Spain

Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 459-463.

Résumé
Abstract

Nous considérons l'équation de Schrödinger non-linéaire associée à un potentiel singulier de la forme $a {| u |}^{- (1 - m)} u + b u$ , avec $m \in (0, 1)$ , sur un domaine éventuellement non borné. Nous employons des méthodes d'énergie appropriées pour montrer que si $Re (a) + Im (a) > 0$ et si les données (initiale et source) ont un support compact alors toute solution doit également avoir un support compact pour tout $t > 0$ . Cette propriété contraste avec le comportement des solutions associées aux potentiels réguliers $(m ⩾ 1)$ . Des résultats similaires sont également établis pour le problème stationnaire associé et pour les solutions auto-similaires sur l'espace entier et le potentiel $a {| u |}^{- (1 - m)} u$ . L'existence des solutions est obtenue par des méthodes de compacité sous certaines conditions.

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a {| u |}^{- (1 - m)} u + b u$ , for some $m \in (0, 1)$ , on a possible unbounded domain. We use some suitable energy methods to prove that if $Re (a) + Im (a) > 0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t > 0$ . This property contrasts with the behavior of solutions associated to regular potentials $(m ⩾ 1)$ . Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a {| u |}^{- (1 - m)} u$ . The existence of solutions is obtained by some compactness methods under additional conditions.

Reçu le : 2006-01-20
Publié le : 2006-03-06

DOI : 10.1016/j.crma.2006.01.027

Affiliations des auteurs :

Bégout, Pascal ¹ ; Díaz, Jesús Ildefonso ²

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     author = {B\'egout, Pascal and D{\'\i}az, Jes\'us Ildefonso},
     title = {On a nonlinear {Schr\"odinger} equation with a localizing effect},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--463},
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     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.01.027/}
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Bégout, Pascal; Díaz, Jesús Ildefonso. On a nonlinear Schrödinger equation with a localizing effect. Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 459-463. doi : 10.1016/j.crma.2006.01.027. http://www.numdam.org/articles/10.1016/j.crma.2006.01.027/

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