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 Schrödinger equations by finding solutions to the singular system -Δu + λ1u + u/|x|2 = μ1u3 + βuv2, x ∈ ℝ2, -Δv + λ2v + v/|x|2 = μ2v3 + βu2v, x ∈ ℝ2, u, v ≥ 0, x ∈ ℝ2, where λ1, λ2, μ1, μ2 are positive parameters, β ≠ 0. We show that this system has a positive least energy solutions for the cases when either β is negative or β is positive and small or large. Moreover, if λ1 = λ2, then the solution is unique. We also study the limiting behavior of the least energy solutions in the repulsive case for β → -∞, and phase separation.