Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

Abstract

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} Δu=λ\mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 π\sum_{s=1}^{N} δ_{p_{s}} \quad , \end{equation*} where $\lambda >0$, ${\delta }_{p_s}$ is the Dirac mass at the vetex $p_s$, and $p_1,p_2,\dots ,p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial value $\hat{\lambda }$ such that when $\lambda >\hat{\lambda }$, the generalized Chern-Simons equation has at least two solutions, when $\lambda =\hat{\lambda }$, the generalized Chern-Simons equation has a solution, and when $\lambda <\hat{\lambda }$, the generalized Chern-Simons equation has no solution.