Partial Differential Equations
On a nonlinear Schrödinger equation with a localizing effect
Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 459-463.

We consider the nonlinear Schrödinger equation associated to a singular potential of the form a|u|(1m)u+bu, for some m(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 (m1). Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential a|u|(1m)u. The existence of solutions is obtained by some compactness methods under additional conditions.

Nous considérons l'équation de Schrödinger non-linéaire associée à un potentiel singulier de la forme a|u|(1m)u+bu, avec m(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 (m1). 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|(1m)u. L'existence des solutions est obtenue par des méthodes de compacité sous certaines conditions.

Received:
Published online:
DOI: 10.1016/j.crma.2006.01.027
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
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Bégout, Pascal; Díaz, Jesús Ildefonso. On a nonlinear Schrödinger equation with a localizing effect. Comptes Rendus. Mathématique, Volume 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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