On Non-topological Solutions of the ${\mathbf{G}}_2$ Chern-Simons System

Abstract

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form: \begin{equation}\label{e001} \left\{\begin{array}{c} Δu_1+(\sum_{i=1}^2K_{1i}e^{u_i} -\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{1i}e^{u_j}K_{ij})=4π\displaystyle \sum_{j=1}^{N_1}δ_{p_j}\\ Δu_2+ (\sum_{i=1}^2K_{2i}e^{u_i}-\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{2i}e^{u_j}K_{ij})=4π\displaystyle \sum_{j=1}^{N_2}δ_{q_j} \end{array} \right.\mbox{in}\; \mathbb{R}^2, \end{equation} where $K$ is the Cartan matrix of rank $2$. There are three Cartan matrix of rank 2: ${\mathbf{A}}_2$, ${\mathbf{B}}_2$ and ${\mathbf{G}}_2$. A long-standing open problem for \eqref{e001} is the question of the existence of non-topological solutions. In a previous paper \cite{ALW}, we have proven the existence of non-topological solutions for the ${\mathbf{A}}_2$ and ${\mathbf{B}}_2$ Chern-Simons system. In this paper, we continue to consider the ${\mathbf{G}}_2$ case. We prove the existence of non-topological solutions under the condition that either $N_2\sum _{j=1}^{N_1}{p}_j=N_1\sum _{j=1}^{N_2}{q}_j$ or $N_2\sum _{j=1}^{N_1}{p}_j\ne N_1\sum _{j=1}^{N_2}{q}_j$ and $N_1,N_2>1$, $|N_1-N_2|\ne 1$. We solve this problem by a perturbation from the corresponding ${\mathbf{G}}_2$ Toda system with one singular source. Combining with \cite{ALW}, we have proved the existence of non-topological solutions to the Chern-Simons system with Cartan matrix of rank $2$.

40 pages