Abstract
This paper is a sequel to the paper (Hamano and Ikeda, Proc. Amer. Math. Soc. To appear). In this paper, we consider the nonlinear Schrödinger equation with a real-valued linear potential and p-th order gauge invariant nonlinearity. Our aim of this paper is to characterize the ground state of the elliptic equation corresponding to the time-dependent problem by the virial functional (see Definition 1.2). In order to study the ground state, we consider attainability of a minimization problem about the elliptic equation. As its application, we give a sufficient condition on the initial data, under which the energy solution can be extended globally in time. We note that the condition is defined by the minimum value of the minimization problem.
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Acknowledgements
The authors would like to express deep appreciation to Professor Takahisa Inui and Professor Masahito Ohta. Professor Masahito Ohta introduced this problem to the authors and Professor Takahisa Inui gave a lot of useful comments. The first author is supported by the Grant-in-Aid for Japan Society for the Promotion of Science (No.19J13300). The second author is 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.
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Hamano, M., Ikeda, M. (2020). Characterization of the Ground State to the Intercritical NLS with a Linear Potential by the Virial Functional. In: Georgiev, V., Ozawa, T., Ruzhansky, M., Wirth, J. (eds) Advances in Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-58215-9_12
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DOI: https://doi.org/10.1007/978-3-030-58215-9_12
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