Doubly periodic vortices for a Chern-Simons model

In this paper we establish the existence of vortex solutions for a relativistic self-dual U(1)×U(1) Chern-Simons system over a doubly periodic domain. We use a constrained minimization approach to obtain the existence result, the system admits a solution when the Chern-Simons coupling parameter κ>0 is sufficiently small; while no solution exists when κ>0 is not sufficiently small. And we also get the quantized integral.