Self-dual radial non-topological solutions to a competitive Chern-Simons model

We study a non-Abelian Chern-Simons system of rank 2:(Δu1Δu2)+K(eu1eu2)−K(eu100eu2)K(eu1eu2)=(4πN1δ04πN2δ0)in R2, where N1,N2∈N∪{0}, δ0 is the Dirac measure at 0, and K=(aij) is a 2×2 matrix satisfying a11,a22>0, a12,a21<0 and det⁡K>0, including the Cartan matrix B2. The existence of non-topological solutions has remained a long-standing open problem. Here by applying the degree theory, we prove the existence of radial non-topological solutions (u1,u2) satisfying the prescribed asymptotic condition uk(x)=−2αkln⁡|x|+O(1) as |x|→∞ for some αk>1. We also construct bubbling solutions to show that the range of (α1,α2) is optimal in some sense. This generalizes a recent work by Choe, Kim and the second author, where the SU(3) case (i.e. K is the Cartan matrix A2) was investigated.