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Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials

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Abstract

We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type

$$-{\varepsilon}^2\Delta u + Vu = u^p\quad{{\rm in}\,{\mathbb R}^N},$$

where N ≥ 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and \({\frac{N}{N-2} < p < \frac{N+2}{N-2}}\) , then for small ε the problem admits positive solutions which concentrate as ε → 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, UK

    Vitaly Moroz

  2. Département de Mathématique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Jean Van Schaftingen

Authors
  1. Vitaly Moroz
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  2. Jean Van Schaftingen
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Correspondence to Vitaly Moroz.

Additional information

Communicated by A. Malchiodi.

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Moroz, V., Van Schaftingen, J. Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials. Calc. Var. 37, 1–27 (2010). https://doi.org/10.1007/s00526-009-0249-y

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  • DOI: https://doi.org/10.1007/s00526-009-0249-y

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