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Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima

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Abstract

We study positive bound states for the equation \({- \varepsilon^2 \Delta u + Vu = u^p, \quad {\rm in} \quad \mathbb{R}^N}\), where \({\varepsilon > 0}\) is a real parameter, \({\frac{N}{N-2} < p < \frac{N+2}{N-2}}\) and V is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential V without any restriction on the potential.

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Authors and Affiliations

  1. Université Catholique de Louvain, Institut de Recherche en Mathématique et en Physique, Chemin du Cyclotron 2, bte L7.01.01, 1348, Louvain-la-Neuve, Belgium

    Jonathan Di Cosmo & Jean Van Schaftingen

  2. Département de Mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, 1050, Bruxelles, Belgium

    Jonathan Di Cosmo

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  1. Jonathan Di Cosmo
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  2. Jean Van Schaftingen
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Correspondence to Jean Van Schaftingen.

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Communicated by A. Malchiodi.

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Di Cosmo, J., Van Schaftingen, J. Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima. Calc. Var. 47, 243–271 (2013). https://doi.org/10.1007/s00526-012-0518-z

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  • DOI: https://doi.org/10.1007/s00526-012-0518-z

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