Existence of Radial Mixed Type Solutions in Chern–Simons Theories of Rank 2 in $$\mathbb {R}^2$$

Abstract

We are concerned with Chern–Simons gauge theories of rank 2 in $$\mathbb {R}^2$$ such as SU(3), SO(5), and $$G_2$$ Chern–Simons theory. One can reduce the self-dual equations of these models to a $$2\times 2$$ system of elliptic equations. There exist three types of solutions of these elliptic system classified according to their asymptotic behaviors, namely, topological, nontopological, and mixed type solutions. In this paper, we consider the existence of radial mixed type solutions of these models. By counting the degree of the corresponding operator, we show the existence under a reasonable condition. Our result on the SU(3) Chern–Simons theory is an improvement of the existence result in Choe et al. (2017). Also, our results for the SO(5) and $$G_2$$ Chern–Simons theories are entirely new, to which the variational approach developed in Choe et al. (2017) cannot be applied.