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We consider non-topological solutions of a nonlinear elliptic system problemderived from the $SU(3)$ Chern-Simons models in $\mathbb{R}^2$. The existenceof non-topological solutions even for radial symmetric case has been a longstanding open problem. Recently, [Choe, Kim, Lin (2015, 2016)] showed theexistence of radial symmetric non-topological solution when the vortex pointscollapse. However, the arguments in [Choe, Kim, Lin (2015, 2016)] cannot workfor an arbitrary configuration of vortex points. In this paper, we develop anew approach by using different scalings for different components of the systemto construct a family of non-topological solutions, which blows up at infinity.

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