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

Abstract

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} \Delta u &=\lambda\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)+4\pi\sum\limits_{j=1}^{k_1}m_j\delta_{p_j},\\ \Delta \upsilon&=\lambda\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)+4\pi\sum\limits_{j=1}^{k_2}n_j\delta_{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 $$\left\{\begin{aligned} \Delta u &=\lambda\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1) +4\pi\sum\limits_{j=1}^{k_1}m_j\delta_{p_j},\\ \Delta \upsilon&=\lambda\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4\pi\sum\limits_{j=1}^{k_2}n_j\delta_{q_j}, \end{aligned} \right. $$ and the classical Chern-Simons equation $$ \Delta u=\lambda \mathrm{e}^u(\mathrm{e}^u-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j}.$$