Existence theorems for a generalized Chern-Simons equation on finite graphs

Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$\Delta u=\lambda \mathrm{e}^u(\mathrm{e}^{bu}-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j}$$ on $G$, where $\lambda$ and $b$ are positive constants; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $\delta_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value $\lambda_c$ such that the equation has a solution if $\lambda\geq \lambda_c$ and the equation has no solution if $\lambda<\lambda_c$. We also prove that if $\lambda>\lambda_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution.