Vortex Condensates for Relativistic Abelian Chern-Simons Model with Two Higgs Scalar Fields and Two Gauge Fields on a Torus

We prove the existence of maximal condensates for the relativistic Abelian Chern-Simons equations involving two Higgs particles and two gauge fields on a torus. After a change of variable, we obtain a variational formulation of the problem whose critical points are equivalent to the original system of the equation. We prove existence of a local minimizer for this functional as well as the existence of a second mountain-pass critical point.