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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (4): 521-528.doi: https://doi.org/10.1007/s10483-010-0412-7

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Nonlinear Schr¨odinger equation with combined power-type nonlinearities and harmonic potential

徐润章1 徐闯2   

  1. 1. College of Science, Harbin Engineering University, Harbin 150001, P. R. China;
    2. Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
  • 收稿日期:2009-07-08 修回日期:2010-03-10 出版日期:2010-04-20 发布日期:2010-04-01

Nonlinear Schr¨odinger equation with combined power-type nonlinearities and harmonic potential

XU Run-Zhang1, XU Chuang2   

  1. 1. College of Science, Harbin Engineering University, Harbin 150001, P. R. China;
    2. Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
  • Received:2009-07-08 Revised:2010-03-10 Online:2010-04-20 Published:2010-04-01

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摘要： This paper discusses a class of nonlinear Schr¨odinger 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.

Abstract: This paper discusses a class of nonlinear Schr¨odinger 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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