A system of elliptic equations arising in Chern–Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern–Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R2 limit from the bounded-domain solutions to obtain a topological solution in R2.