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Global well-posedness below the ground state for the nonlinear Schrödinger equation with a linear potential


Authors: Masaru Hamano and Masahiro Ikeda
Journal: Proc. Amer. Math. Soc. 148 (2020), 5193-5207
MSC (2010): Primary 35Q55, 35J60
DOI: https://doi.org/10.1090/proc/15161
Published electronically: September 17, 2020
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Abstract: In this paper, we study existence of a standing wave solution for the nonlinear Schrödinger equation with a real-valued linear potential in energy-subcritical. Moreover, we also prove global well-posedness to the Cauchy problem with the initial data, whose action is less than that of the ground state with the potential. Hong [Commun. Pure Appl. Anal. 15 (2016), pp. 1571-1601] and the authors showed a scattering result for the problem with the initial data, whose action is less than that of the ground state without the potential. It was noted that the action of the ground state with the potential is greater than that of the ground state without the potential. Our new contribution enables us to treat initial data, which were not treated in those papers.


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Additional Information

Masaru Hamano
Affiliation: Department of Mathematics, Graduate School of Science and Engineering Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, Saitama 338-8570, Japan
ORCID: 0000-0001-5057-3614
Email: m.hamano.733@ms.saitama-u.ac.jp

Masahiro Ikeda
Affiliation: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan; and Center for Advanced Intelligence Project, Riken, Japan
MR Author ID: 940764
Email: masahiro.ikeda@keio.jp/masahiro.ikeda@riken.jp

DOI: https://doi.org/10.1090/proc/15161
Keywords: Nonlinear Schr\"odinger equation, standing wave, global well-posedness
Received by editor(s): November 1, 2019
Received by editor(s) in revised form: November 15, 2019, and April 11, 2020
Published electronically: September 17, 2020
Additional Notes: The first author was supported by Grant-in-Aid for Japan Society for the Promotion of Science (No.19J13300).
The second author was supported by the Grant-in-Aid for Scientific Research (B) (No.18H01132), Young Scientists Research (No.19K14581), Japan Society for the Promotion of Science and JST CREST (JPMJCR1913), Japan.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2020 American Mathematical Society