Existence of non-topological solutions for a skew-symmetric Chern-Simons system

We investigate the existence of non-topological solutions $(u_1,u_2)$ satisfying $u_{i}(x)=-2β_i\ln|x|+O(1),\quad\text{as }|x|\rightarrow +\infty,$ such that $β_i>1$ and $(β_1-1)(β_2-1)>(N_1+1)(N_2+1),$ for a skew-symmetric Chern-Simons system. By the bubbling analysis and the Leray-Schauder degree theory, we get the existence results except for a finite set of curves: $\frac{N_1}{β_1+N_1}+\frac{N_2}{β_2+N_2}=\frac{k-1}{k},k=2,\cdots,\max(N_1,N_2).$ This generalizes a previous work by Choe-Kim-Lin \cite{ChoeKimLin2011}.

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