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: $$\left{\begin{aligned} \Delta u &amp;=\lambda\mathrm{e}^{{\upsilon}(\mathrm{e}^{{u}-1)+4\pi\sum\limits_{j=1}^{{k_1}m_j\delta_{p_j},}}} \Delta \upsilon&amp;=\lambda\mathrm{e}^{{u}(\mathrm{e}^{{\upsilon}-1)+4\pi\sum\limits_{j=1}^{{k_2}n_j\delta_{q_j},}}} \end{aligned} \right. $$ here, $\lambda\in\mathbb{R}<em>+$, $k_1$ 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 $\delta</em>{p}$ denotes the Dirac mass at vertex $p$. Write $$g=4\pi\sum_{j=1}^{{k_1}m_j\delta_{p_j},\} h=4\pi\sum_{j=1}^{{k_2}n_j\delta_{q_j},\} B = 4\pi\sum_{j=1}^{k_1}m_j + 4\pi\sum_{j=1}<sup>{k_2}n_j.$$ 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 $\mathbb{Z}<sup>n$ is large enough or $\lambda$ is large enough.