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On Nonlinear Schrödinger Equations with Repulsive Inverse-Power Potentials

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Abstract

In this paper, we consider the Cauchy problem for nonlinear Schrödinger equations with repulsive inverse-power potentials

$$ i \partial _{t} u + \Delta u - c |x|^{-\sigma } u = \pm |u|^{\alpha }u, \quad c>0. $$

We study the local and global well-posedness, finite time blow-up and scattering in the energy space for the equation. These results extend a recent work of Miao-Zhang-Zheng (2018, arXiv:1809.06685) to a general class of inverse-power potentials and higher dimensions.

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Acknowledgement

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The author would like to express his deep gratitude to his wife - Uyen Cong for her encouragement and support. He would like to thank Prof. Changxing Miao for fruitful discussions on the energy scattering. He also would like to thank the reviewer for his/her helpful comments and suggestions.

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  1. Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655, Villeneuve d’Ascq Cedex, France

    Van Duong Dinh

  2. Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

    Van Duong Dinh

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  1. Van Duong Dinh
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Correspondence to Van Duong Dinh.

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Dinh, V.D. On Nonlinear Schrödinger Equations with Repulsive Inverse-Power Potentials. Acta Appl Math 171, 14 (2021). https://doi.org/10.1007/s10440-020-00382-2

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