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} 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 $$\{\begin{array}{rl}\Delta u& =\lambda {\mathrm{e}}^{\upsilon }({\mathrm{e}}^u-1)+4\pi \sum _{j=1}^{k_1}{m}_j{\delta }_{p_j},\\ \Delta \upsilon & =\lambda {\mathrm{e}}^u({\mathrm{e}}^{\upsilon }-1)+4\pi \sum _{j=1}^{k_2}{n}_j{\delta }_{q_j},\end{array}$$ and the classical Chern-Simons equation $$\Delta u=\lambda {\mathrm{e}}^u({\mathrm{e}}^u-1)+4\pi \sum _{j=1}^N{\delta }_{p_j}.$$

13 pages