Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs

Abstract

In this paper, we consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\left\{\begin{aligned} \Delta u &=\lambda\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1)+4\pi\sum\limits_{j=1}^{k_1}m_j\delta_{p_j}, \Delta \upsilon&=\lambda\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4\pi\sum\limits_{j=1}^{k_2}n_j\delta_{q_j}, \end{aligned} \right. $$ here, $\lambda\in\mathbb{R}_+$, $k_1$ and $k_2$ are two positive integers, $m_j\in\mathbb{N}\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j\in\mathbb{N}\,(j=1,2,\cdot\cdot\cdot,k_2)$, and $\delta_{p}$ denotes the Dirac mass at vertex $p$. Write $$g=4\pi\sum_{j=1}^{k_1}m_j\delta_{p_j},\ h=4\pi\sum_{j=1}^{k_2}n_j\delta_{q_j},\ B = 4\pi\sum_{j=1}^{k_1}m_j + 4\pi\sum_{j=1}^{k_2}n_j.$$ For any fixed $g,h$, we prove the existence of the topological solutions to the systems, then obtain the asymptotic behaviors of topological solutions as $\lambda \rightarrow 0_+$ and $\lambda \rightarrow +\infty$, and finally prove the uniqueness of the topological solutions when the dimension of lattice graph $\mathbb{Z}^n$ is large enough or $\lambda$ is large enough.