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Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

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Abstract

This paper discusses a class of nonlinear Schrödinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.

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Authors and Affiliations

  1. College of Science, Harbin Engineering University, Harbin, 150001, P. R. China

    Run-zhang Xu  (徐润章)

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P. R. China

    Chuang Xu  (徐 闯)

Authors
  1. Run-zhang Xu  (徐润章)
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  2. Chuang Xu  (徐 闯)
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Correspondence to Run-zhang Xu  (徐润章).

Additional information

Communicated by Li-qun CHEN

Project supported by the National Natural Science Foundation of China (Nos. 10871055 and 10926149), the Natural Science Foundation of Heilongjiang Province (Nos. A200702 and A200810), the Science and Technology Foundation of Education Office of Heilongjiang Province (No. 11541276), and the Foundational Science Foundation of Harbin Engineering University

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Xu, Rz., Xu, C. Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential. Appl. Math. Mech.-Engl. Ed. 31, 521–528 (2010). https://doi.org/10.1007/s10483-010-0412-7

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  • DOI: https://doi.org/10.1007/s10483-010-0412-7

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