Classification of the entire radial self-dual solutions to non-Abelian Chern–Simons systems

Abstract

We consider the entire radial solutions to the non-Abelian Chern–Simons systems of rank 2(ΔuΔv)=−K(euev)+K(eu00ev)K(euev)+(4πN1δ04πN2δ0)(0.1)in R2, where Ni⩾0, i=1,2 and K=(aij) is a 2×2 matrix satisfying a11, a22>0, a12, a21<0 and a11a22−a12a21>0. This system is motivated by the relativistic non-Abelian Chern–Simons model, Lozano–Marqués–Moreno–Schaposnik model of bosonic sector of N=2 supersymmetric Chern–Simons–Higgs theory, and Gudnason model of N=2 supersymmetric Yang–Mills–Chern–Simons–Higgs theory. Understanding the structure of entire radial solutions is one of fundamental issues for the system of nonlinear equations. We prove that any entire radial solutions of (0.1) must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. As an application of this classification, we prove that the two components u and v have intersection at most finite times.