Comptes Rendus
Partial Differential Equations
Nonlinear Schrödinger equations with potentials vanishing at infinity
Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 903-908.

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

ε2Δu+V(x)u=K(x)up,xRN,
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 ε0. We also investigate the qualitative properties of the solution as ε0.

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

ε2Δu+V(x)u=K(x)up,xRN,
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 ε0. Nous étudions également les propriétés qualitatives de cette solution lorsque ε0.

Received:
Published online:
DOI: 10.1016/j.crma.2006.04.011

Denis Bonheure 1; Jean Van Schaftingen 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
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Denis Bonheure; Jean Van Schaftingen. 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://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.011/

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