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 $λ>0$, $b>a>0$, $m_j>0\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j>0\,(j=1,2,\cdot\cdot\cdot,k_2)$, $δ_{p}$ 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 $λ$ goes to infinity. This method is also applicable to the Chern-Simons system $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1) +4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ and the classical Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^u-1)+4π\sum\limits_{j=1}^{N}δ_{p_j}.$$

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4