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Nonlinear Schrödinger equations with potentials vanishing at infinity

par Bonheure, Denis ;Van Schaftingen, Jean
Référence Comptes rendus. Mathématique, 342, 12, page (903-908)
Publication Publié, 2006

Article révisé par les pairs

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Résumé :

In this Note, we deal with stationary nonlinear Schrödinger equations of the form- ε2 Δ u + 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. To cite this article: D. Bonheure, J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 342 (2006). © 2006 Académie des sciences.