Blow up at infinity in the SU(3) Chern-Simons model, part I

We consider non-topological solutions of a nonlinear elliptic system problem (see (1.4) below) derived from the SU(3) Chern-Simons models in R². The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, Choe, Kim, and Lin in [7,8] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [7,8] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.