# Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $\mathbb{R}^2$

## TL;DR

This paper proves the existence of mixed type solutions in non-Abelian Chern-Simons gauge theories of rank 2 in -dimensional space, addressing a long-standing open problem by analyzing blow-up behaviors and mass contributions.

## Contribution

It establishes the existence of mixed type solutions with arbitrary vortex configurations in non-Abelian Chern-Simons models, using novel scaling and blow-up analysis techniques.

## Key findings

- Existence of mixed solutions for arbitrary vortex points.
- Identification of conditions where a priori bounds fail.
- Development of methods to control mass contributions at infinity.

## Abstract

We consider the Chern-Simons gauge theory of rank $2$ such as $SU(3)$, $SO(5)$, and $G_2$ Chern-Simons model in $\mathbb{R}^2$. There may exist three types of solutions in these theories, that is, topological, nontopological, and mixed type solutions. Among others, mixed type solutions can only exist in non-Abelian Chern-Simons models. We show the existence of mixed type solutions with an arbitrary configuration of vortex points which has been a long-standing open problem. To show it, as the first step, we need to find when a priori bound would fail. For the purpose, we shall find partially blowing up mixed type solutions by using different scalings for different components. Due to the different scalings, we should control the mass contribution from infinity which is one of the important parts in this paper.