耦合非线性Schrdinger方程组的Neumann问题

Neumann Problem for Coupled Nonlinear Schrdinger Equations

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 TaiChia和WeiJuncheng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.

In this paper,we consider existence and concentration phenomena of least energy solutions of coupled nonlinear Schr（o｜¨）dinger 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.