Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs

In this paper, we investigate a system of equations derived from the $$\text {U}(1)\times \text {U}(1)$$ Abelian Chern-Simons model: $$\begin{aligned}\left\{ \begin{aligned} \Delta u&=\lambda \left( a(b-a)\textrm{e}^u-b(b-a)\textrm{e}^{\upsilon }+a^2\textrm{e}^{2u}-ab\textrm{e}^{2\upsilon }+b(b-a)\textrm{e}^{u+\upsilon } \right) +4\pi \sum \limits _{j=1}^{k_1}m_j\delta _{p_j},\\ \Delta \upsilon&=\lambda \left( -b(b-a)\textrm{e}^u+a(b-a)\textrm{e}^{\upsilon }-ab\textrm{e}^{2u}+a^2\textrm{e}^{2\upsilon }+b(b-a)\textrm{e}^{u+\upsilon } \right) +4\pi \sum \limits _{j=1}^{k_2}n_j\delta _{q_j}, \end{aligned} \right. \end{aligned}$$ on finite graphs. Here, $$\lambda >0$$ , $$b>a>0$$ , $$m_j>0\, (j=1,2,\ldots ,k_1)$$ , $$n_j>0\,(j=1,2,\ldots ,k_2)$$ , and $$\delta _{p}$$ denotes the Dirac delta mass at vertex p. We establish an iteration scheme and prove the existence of solutions. Additionally, we propose a novel method to derive the asymptotic behavior of solutions as $$\lambda $$ approaches infinity. This method is also applicable to the Chern-Simons system: $$\begin{aligned} \left\{ \begin{aligned} \Delta u&=\lambda \textrm{e}^{\upsilon }(\textrm{e}^{u}-1)+4\pi \sum \limits _{j=1}^{k_1}m_j\delta _{p_j},\\ \Delta \upsilon&=\lambda \textrm{e}^{u}(\textrm{e}^{\upsilon }-1)+4\pi \sum \limits _{j=1}^{k_2}n_j\delta _{q_j}, \end{aligned} \right. \end{aligned}$$ and the classical Chern-Simons equation: $$\begin{aligned} \Delta u=\lambda \textrm{e}^u(\textrm{e}^u-1)+4\pi \sum \limits _{j=1}^{N}\delta _{p_j}. \end{aligned}$$