Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials

Vitaly Moroz; Jean Van Schaftingen

doi:10.1016/j.crma.2009.05.009

Partial Differential Equations

Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials
[Existence et concentration pour l'équation de Schrödinger non linéaire avec des potentiels à décroissance rapide]

Présenté par : Haïm Brezis

Vitaly Moroz ¹ ; Jean Van Schaftingen ²

¹ Swansea University, Department of Mathematics, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
² Université Catholique de Louvain, département de mathématique, chemin du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 921-926.

Résumé
Abstract

Nous prouvons l'existence de solutions positives non triviales de

- ε^{2} Δ u + V u = u^{p} dans R^{N},

pour ε petit, où

N ⩾ 3

, V est potentiel continu positif qui a un minimum local strictement positif et

\frac{N}{N - 2} < p < \frac{N + 2}{N - 2}

. Nous n'imposons aucune restriction sur le taux de décroisance de V. En particulier, nous couvrons le cas où le support de V est compact.

The existence of positive solutions to

- ε^{2} Δ u + V u = u^{p} in R^{N},

is proved for small ε, where

N ⩾ 3

, V is a nonnegative continuous potential which has a positive local minimum, and

\frac{N}{N - 2} < p < \frac{N + 2}{N - 2}

. No restriction is imposed on the rate of decay of V; this includes in particular the case where V is compactly supported.

Reçu le : 2009-04-06
Accepté le : 2009-05-11
Publié le : 2009-06-12

DOI : 10.1016/j.crma.2009.05.009

Affiliations des auteurs :

Vitaly Moroz ¹ ; Jean Van Schaftingen ²

¹ Swansea University, Department of Mathematics, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
² Université Catholique de Louvain, département de mathématique, chemin du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

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     author = {Vitaly Moroz and Jean Van Schaftingen},
     title = {Existence and concentration for nonlinear {Schr\"odinger} equations with fast decaying potentials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {921--926},
     publisher = {Elsevier},
     volume = {347},
     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.05.009},
     language = {en},
}

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Vitaly Moroz; Jean Van Schaftingen. Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 921-926. doi : 10.1016/j.crma.2009.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.009/

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