On non-topological solutions of the $\mathbf{G}_2$ Chern–Simons system

Juncheng Wei (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada; and Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Abstract

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form:\[(0,1)\begin{cases}\displaystyle \Delta u_1 + \left (\sum_{i=1}^{2} K_{1i} e^{u_i} - \sum_{i=1}^{2} \sum_{j=1}^{2} e^{u_i} K_{1i} e^{u_j} K_{ij} \right ) = 4 \pi \sum_{j=1}^{N_1} \delta_{p_j} \\\displaystyle \Delta u_2 + \left (\sum_{i=1}^{2} K_{2i} e^{u_i} - \sum_{i=1}^{2} \sum_{j=1}^{2} e^{u_i} K_{2i} e^{u_j} K_{ij} \right ) = 4 \pi \sum_{j=1}^{N_2} \delta_{p_j}\end{cases}\; \; \textrm{ in } \mathbb{R}^2 \: \textrm{,}\]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 (0.1) is the question of the existence of non-topological solutions. In a previous paper [1], we have proved 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 $N_2 \Sigma^{N_1}_{j=1} \: p_j = N_1 \Sigma^{N_2}_{j=1} \: q_j$ or $ N_2 \Sigma^{N_1}_{j=1} \: p_j \neq N_1 \Sigma^{N_2}_{j=1} \: q_j $ and $N_1 , N_2 \gt 1, \lvert N_1 - N_2 \rvert \neq 1$ We solve this problem by a perturbation from the corresponding $\mathbf{G}_2$ Toda system with one singular source. Combining with [1], we have proved the existence of nontopological solutions to the Chern–Simons system with Cartan matrix of rank $2$.

2010 Mathematics Subject Classification

Primary 35J60. Secondary 35B10, 58J37.

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Published 3 November 2016