Existence theorems for non-Abelian Chern-Simons-Higgs vortices with flavor

In this paper we establish the existence of vortex solutions for a Chern-Simons-Higgs model with gauge group SU(N)×U(1) and flavor SU(N). These symmetries ensure the existence of genuine non-Abelian vortices through a color-flavor locking. Under a suitable ansatz we reduce the problem to a 2×2 system of nonlinear elliptic equations with exponential terms. We study this system over the full plane and over a doubly periodic domain, respectively. For the planar case we use a variational argument to establish the existence result and derive the decay estimates of the solutions. Over the doubly periodic domain we show that the system admits at least two gauge-distinct solutions carrying the same physical energy by using a constrained minimization approach and the mountain-pass theorem. In both cases we get the quantized vortex magnetic fluxes and electric charges.