Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential

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. For more information see https://ejde.math.txstate.edu/Volumes/2020/108/abstr.html