Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 108, pp. 1-20.

Title: Positive vortex solutions and phase separation for coupled Schrodinger
       system with singular potential
       
Authors: Jin Deng    (Jiangxi Normal Univ., Nanchang, Jiangxi, China)
         Aliang Xia  (Jiangxi Normal Univ., Nanchang, Jiangxi, China)
         Jianfu Yang (Jiangxi Normal Univ., Nanchang, Jiangxi, China)

Abstract:
 We consider the existence of rotating solitary waves (vortices) for a coupled
 Schrodinger equations by finding solutions to the singular system
 $$\displaylines{
 -\Delta u+\lambda_1 u+\frac{u}{|x|^2}=\mu_1 u^3+\beta u v^2, \quad x\in\mathbb{R}^2, \cr
 -\Delta v+\lambda_2 v+\frac{v}{|x|^2}=\mu_2 v^3+\beta u^2 v, \quad x\in\mathbb{R}^2, \cr
 u,v \geq 0,\quad x\in\mathbb{R}^2,
 }$$
 where $\lambda_1,\lambda_2,\mu_1, \mu_2$ are positive parameters, $\beta\neq 0$.
 We show that this system has a positive least energy solution for the cases 
 When either $\beta$ is negative or 
\beta
 is positive and  small or large.
 Moreover, if $\lambda_1=\lambda_2$, then the solution is unique.
 We also study the limiting behavior of the least energy solutions in the
 repulsive case for $\beta\to-\infty$, and phase separation.

 
Submitted March 5, 2020. Published October 30, 2020.
Math Subject Classifications: 35J20, 35B08, 35B40.
Key Words: Schodinger equation; singular potential; Nehari manifold.