耦合Schrodinger系统变号解的渐进行为

We study theasymptotic behavior of nodal solution for the coupled nonlinear Schr"{o}dinger system e egin{cases}- arepsilon^2Deltau+lambda_1u=mu_1u^3+ eta uv^2,quad &xin Omega, onumber\- arepsilon^2Delta v+lambda_2v=mu_2v^3+ eta vu^2,quad &xinOmega, onumber\u=v=0,quad &xinpartialOmega, onumberend{cases} ee where $Omegasubset mathbb{R}^N$$(Nleq 3)$ is a smooth and bounded domain, $lambda_1,$$lambda_2,$ $mu_1$, $ mu_2>0$ and $ eta<0$. It is shown that the semi-nodal solution $(u, v)$, that is, one component changes sign and another one is positive, has exactly onepositive and one negative peaks, which converge to two distinct points of $Omega$ as $ arepsilon ightarrow 0,$ and another component has exactly onepositive peak, which converge to a point of $Omega$ as$ arepsilon ightarrow 0.$ We also prove the distances between the local maximum points, the local minimum points and the boundary of $Omega$ divided by $ arepsilon$ diverge.