数学物理学报 ›› 2009, Vol. 29 ›› Issue (5): 1398-1414.

• 论文 • 上一篇    下一篇

耦合非线性Schrödinger方程组的Neumann问题

  

  1. 上海理工大学田家炳理学院 上海 200093
  • 收稿日期:2007-12-18 修回日期:2009-05-27 出版日期:2009-10-25 发布日期:2009-10-25
  • 基金资助:

    上海市优秀青年教师科研专项基金资助

Neumann Problem for Coupled Nonlinear Schrödinger Equations

  1. Tin Ka-Ping College of Science, University of Shanghai |for |Science and Technology, Shanghai 200093
  • Received:2007-12-18 Revised:2009-05-27 Online:2009-10-25 Published:2009-10-25
  • Supported by:

    上海市优秀青年教师科研专项基金资助

摘要:

该文考虑一类耦合椭圆型非线性Schr\"{o}dinger方程组的Neumann问题极小能量解(基态解)的存在性和集中性质. 主要研究极小能量解的尖点, 即最大值点的位置. 利用  Lin Tai-Chia  和 Wei Juncheng 研究  Dirichlet 问题的方法, 该文首先得到了相应Neumann问题的极小能量解的存在性.  当相当于Planck常数的小参数趋于零时, 该文证明了极小能量解的尖点向定义区域的边界靠近, 并且能量集中在这些尖点处. 另外, 方程组解的两个分支解相互吸引或排斥时, 它们的尖点也相互吸引或排斥.

关键词: 极小能量解的集中, Nehari 流形, 山路引理, 耦合非线性 Schrödinger方程组

Abstract:

In this paper, we consider  existence and concentration phenomena of least energy solutions of coupled  nonlinear Schrödinger systems  with Neumann boundary conditions. The focus  is on  the locations of  peaks (maximum points) of the least energy solutions. Following  Tai-Chia Lin and Juncheng Wei's procedure  for  Dirichlet  problem,  least energy solutions for Neumann problem are obtained. As the small perturbed parameter goes to zero, we prove that the peaks of the least energy solutions  approach to the boundary of domain and the energy concentrates around these peaks. On the other hand, peaks of  the two states attract or repulse each other
depending on the interaction between them  to be   attractive or repulsive.

Key words: Concentration of least energy solutions, Nehari manifold, Mountain pass theorem, Coupled nonlinear Schrödinger system

中图分类号: 

  • 35B25