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Doubly Periodic Self-Dual Vortices for a Relativistic Non-Abelian Chern--Simons Model

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions for a $2\times2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern--Simons vortices. We show that the given system admits at least two solutions when the Chern--Simons coupling parameter $κ>0$ is sufficiently small; while no solutions exist for $κ>0$ sufficiently large. As in [36] we use a variational formulation of the problem. Thus, we obtain a first solution via a (local) minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as $κ\to 0$. Then we obtain the second solution by a min-max procedure of "mountain pass" type.