数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (2): 405-424.doi: 10.1016/S0252-9602(17)30011-5

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BOUND STATES OF SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND VANISHING POTENTIALS

巴娜1, 代晋军2   

  1. 1. School of Science, Hubei University of Technology, Wuhan 430068, China;
    2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • 收稿日期:2016-08-02 出版日期:2017-04-25 发布日期:2017-04-25
  • 作者简介:Na BA,E-mail:bana1002@126.com;Jinjun DAI,E-mail:jjdai@mail.ccnu.edu.cn
  • 基金资助:

    Na Ba was supported by National Natural Science Foundation of China (11201132), Scientific Research Foundation for Ph.D of Hubei University of Technology (BSQD12065), and the Scientific Research Project of Education Department of Hubei Province (Q20151401).

BOUND STATES OF SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND VANISHING POTENTIALS

Na BA1, Jinjun DAI2   

  1. 1. School of Science, Hubei University of Technology, Wuhan 430068, China;
    2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • Received:2016-08-02 Online:2017-04-25 Published:2017-04-25
  • Supported by:

    Na Ba was supported by National Natural Science Foundation of China (11201132), Scientific Research Foundation for Ph.D of Hubei University of Technology (BSQD12065), and the Scientific Research Project of Education Department of Hubei Province (Q20151401).

摘要:

We study the bound states to nonlinear Schrödinger equations with electromagnetic fields ih (∂ψ/∂t)=(h/i▽-A(x))2ψ + V(x)ψ-K(x)|ψ|p-1ψ=0, on R+×RN. Let G (x)=[V (x)](p+1)/(p-1)-N/2[K(x)]-2/(p-1) and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x, t)=e-iEt/h uh(x) with uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V (x) and K(x) are allowed to be either compactly supported or unbounded at infinity.

关键词: Multi-bump solutions, nonlinear Schrödinger equation, electromagnetic fields, potentials compactly supported or unbounded at infinity

Abstract:

We study the bound states to nonlinear Schrödinger equations with electromagnetic fields ih (∂ψ/∂t)=(h/i▽-A(x))2ψ + V(x)ψ-K(x)|ψ|p-1ψ=0, on R+×RN. Let G (x)=[V (x)](p+1)/(p-1)-N/2[K(x)]-2/(p-1) and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x, t)=e-iEt/h uh(x) with uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V (x) and K(x) are allowed to be either compactly supported or unbounded at infinity.

Key words: Multi-bump solutions, nonlinear Schrödinger equation, electromagnetic fields, potentials compactly supported or unbounded at infinity