Partial Differential Equations

Nonlinear Schrödinger equations with potentials vanishing at infinity

[Équations de Schrödinger non linéaires avec potentiels s'annulant à l'infini]

Bonheure, Denis¹ ; 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, Tome 342 (2006) no. 12, pp. 903-908.

Abstract (VO)

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

−ε2Δu+V(x)u=K(x)up,x∈RN,

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∼0. We also investigate the qualitative properties of the solution as ε→0.

Résumé

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

−ε2Δu+V(x)u=K(x)up,x∈RN,

où V,K>0 et 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∼0. Nous étudions également les propriétés qualitatives de cette solution lorsque ε→0.

Détail

Reçu le : 2006-03-30
Publié le : 2006-05-15

DOI : 10.1016/j.crma.2006.04.011

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