A new type of non-topological bubbling solutions to a competitive Chern-Simons model

TL;DR

This paper constructs new radial non-topological bubbling solutions for a non-Abelian Chern-Simons system, revealing different blow-up behaviors and extending previous results to new Lie algebra cases.

Contribution

It introduces a novel type of bubbling solutions for the system, using the shooting method, and generalizes prior work to include the B2 Lie algebra case.

Findings

Existence of new bubbling solutions with partial blow-up in different regions.

Confirmation of differences between A2 and B2 cases.

Extension of previous results to B2 Lie algebra case.

Abstract

We study a non-Abelian Chern-Simons system in $\mathbb{R}^2$, including the simple Lie algebras $A_2$ and $B_2$. In a previous work, we proved the existence of radial non-topological solutions with prescribed asymptotic behaviors via the degree theory. We also constructed a sequence of bubbling solutions with only one component blowing up partially at infinity. In this paper, we construct a sequence of radial non-topological bubbling solutions of another type via the shooting argument. One component of these bubbling solutions locally converge to a non-topological solution of the Chern-Simons-Higgs scalar equation, but both components blow up partially in different regions at infinity at the same time. This generalizes a recent work by Choe, Kim and the second author, where the $SU(3)$ case (i.e. $A_2$) was studied. Our result is new even for the $SU(3)$ case and also confirms the difference between the $SU(3)$ case and the $B_2$ case.