Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a 2 × 2 nonlinear elliptic system arising in the study of self-dual non-Abelian Chern-Simons vortices. We show that the system admits at least two solutions when the Chern-Simons coupling parameter ? >0 sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599-639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as ? ? 0. Then we obtain a second solution by a min-max procedure of "mountain pass" type.