Nonlinear Schrödinger equations with potentials vanishing at infinity

Bonheure, Denis; Van Schaftingen, Jean

doi:10.1016/j.crma.2006.04.011

Partial Differential Equations

Nonlinear Schrödinger equations with potentials vanishing at infinity

Presented by: Haïm Brezis

Bonheure, Denis ¹; Van Schaftingen, Jean ^1,²

¹ Université catholique de Louvain, Institut de Mathématique pure et appliquée, chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium
² Laboratoire d'analyse numérique, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 903-908.

Abstract
Résumé

In this Note, we deal with stationary nonlinear Schrödinger equations of the form

- ε^{2} Δ u + V (x) u = K (x) u^{p}, x \in R^{N},

where

V, K > 0

and

p > 1

is subcritical. We allow the potential V to vanish at infinity and the competing function K to be unbounded. In this framework, positive ground states may not exist. We prove the existence of at least one positive bound state solution in the semi-classical limit, i.e. for

ε \sim 0

. We also investigate the qualitative properties of the solution as

ε \to 0

Dans cette Note, nous considérons des équations de Schrödinger non linéaires stationnaires du type

- ε^{2} Δ u + V (x) u = K (x) u^{p}, x \in R^{N},

où

V, K > 0

p > 1

est sous-critique. Nous considérons un potentiel V qui s'annule éventuellement à l'infini et une fonction de compétition K qui pourrait ne pas être bornée. Dans ce cas, l'existence d'une solution positive d'énergie minimale n'est pas assurée. Nous démontrons l'existence d'au moins une solution positive dans la limite semi-classique, c'est-à-dire pour

ε \sim 0

. Nous étudions également les propriétés qualitatives de cette solution lorsque

ε \to 0

Received: 2006-03-30
Published online: 2006-05-15

DOI: 10.1016/j.crma.2006.04.011

Author's affiliations:

Bonheure, Denis ¹; Van Schaftingen, Jean ^{1, 2}

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     author = {Bonheure, Denis and Van Schaftingen, Jean},
     title = {Nonlinear {Schr\"odinger} equations with potentials vanishing at infinity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {903--908},
     publisher = {Elsevier},
     volume = {342},
     number = {12},
     year = {2006},
     doi = {10.1016/j.crma.2006.04.011},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2006.04.011/}
}

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Bonheure, Denis; Van Schaftingen, Jean. Nonlinear Schrödinger equations with potentials vanishing at infinity. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 903-908. doi : 10.1016/j.crma.2006.04.011. https://www.numdam.org/articles/10.1016/j.crma.2006.04.011/

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