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

In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left{\begin{aligned} Δu &=λ\left(a(b-a)$\mathrm{e}$^u-b(b-a)$\mathrm{e}$^{\upsilon}+a^2$\mathrm{e}$^{2u}-ab$\mathrm{e}$^{2\upsilon}+b(b-a)$\mathrm{e}$^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\ Δ\upsilon&=λ\left(-b(b-a)$\mathrm{e}$^u+a(b-a)$\mathrm{e}$^{\upsilon}-ab$\mathrm{e}$^{2u}+a^2$\mathrm{e}$^{2\upsilon}+b(b-a)$\mathrm{e}$^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. \end{eqnarray*} on finite graphs. Here $\lambda >0$, $b>a>0$, $m_j>0(j=1,2,\cdot \cdot \cdot ,k_1)$, $n_j>0(j=1,2,\cdot \cdot \cdot ,k_2)$, ${\delta }_p$ is the Dirac delta mass at vertex $p$. We establish the iteration scheme and prove existence of solutions. We also develop a new method to get the asymptotic behaviors of solutions as $\lambda$ goes to infinity. This method is also applicable to the Chern-Simons system $$\{\begin{array}{rl}{\displaystyle \mathrm{\Delta }u}& {\displaystyle =\lambda {\mathrm{e}}^{\upsilon }({\mathrm{e}}^u-1)+4\pi \sum _{j=1}^{k_1}{m}_j{\delta }_{p_j},}\\ {\displaystyle \mathrm{\Delta }\upsilon }& {\displaystyle =\lambda {\mathrm{e}}^u({\mathrm{e}}^{\upsilon }-1)+4\pi \sum _{j=1}^{k_2}{n}_j{\delta }_{q_j},}\end{array}$$ and the classical Chern-Simons equation $$\mathrm{\Delta }u=\lambda {\mathrm{e}}^u({\mathrm{e}}^u-1)+4\pi \sum _{j=1}^N{\delta }_{p_j}\mathrm{.}$$