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On concentration of positive bound states of nonlinear Schrödinger equations

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Abstract

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation

$$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$

Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0+; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0+.

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Authors and Affiliations

  1. Department of Mathematics, Tulane University, 70118, New Orleans, LA, USA

    Xuefeng Wang

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  1. Xuefeng Wang
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Communicated by A. Jaffe

Research supported in part by NSF Grant DMS-9105172.

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Wang, X. On concentration of positive bound states of nonlinear Schrödinger equations. Commun.Math. Phys. 153, 229–244 (1993). https://doi.org/10.1007/BF02096642

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  • DOI: https://doi.org/10.1007/BF02096642

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