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

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- by Masaru Hamano and Masahiro Ikeda PDF
- Proc. Amer. Math. Soc.
**148**(2020), 5193-5207 Request permission

## 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.## References

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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
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5193-5207 - MSC (2010): Primary 35Q55, 35J60
- DOI: https://doi.org/10.1090/proc/15161
- MathSciNet review: 4163832