On a nonlinear Schrödinger equation with a localizing effect

Pascal Bégout; Jesús Ildefonso Díaz

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

Pascal Bégout ¹ ; Jesús Ildefonso Díaz 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, Volume 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 :

Pascal Bégout ¹ ; Jesús Ildefonso Díaz 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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     author = {Pascal B\'egout and Jes\'us Ildefonso D{\'\i}az},
     title = {On a nonlinear {Schr\"odinger} equation with a localizing effect},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--463},
     publisher = {Elsevier},
     volume = {342},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.027},
     language = {en},
}

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Pascal Bégout; Jesús Ildefonso Díaz. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.027/

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Anouar Bahrouni; Nizar Bedoui; Hichem Ounaies Multiple solutions with compact support for a quasilinear Schrödinger equation with sign-changing potentials, Complex Variables and Elliptic Equations, Volume 67 (2022) no. 7, pp. 1782-1793 | DOI:10.1080/17476933.2021.1900139 | Zbl:1492.35131
Jesus Ildefonso Diaz; Juan Francisco Padial; Jose Ignacio Tello; Lourdes Tello Complex Ginzburg-Landau equations with a delayed nonlocal perturbation, Electronic Journal of Differential Equations, Volume 2020 (2020) no. 01-132, p. 40 | DOI:10.58997/ejde.2020.40
Jesús Ildefonso Díaz; Juan Francisco Padial; Jose Ignacio Tello; Lourdes Tello Complex Ginzburg-Landau equations with a delayed nonlocal perturbation, Electronic Journal of Differential Equations (EJDE), Volume 2020 (2020), p. 18 (Id/No 40) | Zbl:1441.35223
Nizar Bedoui; Hichem Ounaies Qualitative properties and support compactness of solutions for quasilinear Schrödinger equation with sign changing potentials, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 198 (2020), p. 26 (Id/No 111843) | DOI:10.1016/j.na.2020.111843 | Zbl:1440.35174
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Jesús Ildefonso Díaz On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case, S $\vec{e}$ MA Journal, Volume 74 (2017) no. 3, pp. 255-278 | DOI:10.1007/s40324-017-0115-3 | Zbl:1390.35298
Pascal Bégout; Jesús Ildefonso Díaz A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity, Discrete and Continuous Dynamical Systems, Volume 34 (2014) no. 9, pp. 3371-3382 | DOI:10.3934/dcds.2014.34.3371 | Zbl:1305.35022
Stanislav Antontsev; João-Paulo Dias; Mário Figueira Complex Ginzburg-Landau equation with absorption: existence, uniqueness and localization properties, Journal of Mathematical Fluid Mechanics, Volume 16 (2014) no. 2, pp. 211-223 | DOI:10.1007/s00021-013-0147-0 | Zbl:1291.35205
Pascal Bégout; Jesús Ildefonso Díaz Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – the stationary case, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 29 (2012) no. 1, pp. 35-58 | DOI:10.1016/j.anihpc.2011.09.001 | Zbl:1241.35185

Cité par 9 documents. Sources : Crossref, zbMATH

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