  We study existence and various behaviors of topological multivortices
solutions of the relativistic self-dual Maxwell-Chern-Simons-Higgs system. We
first prove existence of general topological solutions by applying variational
methods to the newly discovered minimizing functional. Then, by an iteration
method we prove existence of topological solutions satisfying some extra
conditions, which we call admissible solutions. We establish asymptotic
exponential decay estimates for these topological solutions. We also
investigate the limiting behaviors of the admissible solutions as parameters in
our system goes to some limits. For the Abelian Higgs limit we obtain strong
convergence result, while for the Chern-Simons limit we only obtained that our
admissible solutions are weakly approximating one of the Chern-Simons
solutions.
