Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in R2

We consider the self-dual equations arising from the Chern-Simons gauge theory of rank 2 such as the SU(3), SO(5), and G2 Chern-Simons model in R2. There are three possible types of solutions in these theories, namely, topological, nontopological, and mixed type solutions. The existence of a mixed type solution for an arbitrary configuration of vortex points has been a long-standing open problem. We construct here a family of mixed type solutions (u1,ε,u2,ε), ε>0 for an arbitrary configuration of vortex points under a mild non-degeneracy condition. The main new idea of our construction of an approximate solution is to use different scalings for different components. In the process of the finite dimensional reduction, all estimates for this particular construction become rather delicate.