Existence of solutions to Chern-Simons-Higgs equations on graphs

Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation $$\Delta u=-\lambda e^{g(u)}\left( e^{g(u)}-1\right)^2+4\pi\sum\limits_{j=1}^{N}\delta_{p_j}$$ on $G$, where $\lambda$ is a positive constant; $g(u)$ is the inverse function of $u=f(\upsilon)=1+\upsilon-e^{\upsilon}$ on $(-\infty, 0]$; $N$ 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 is critical value $\lambda_c$ such that the generalized Chern-Simons-Higgs equation has a solution if and only if $\lambda\geq \lambda_c$ . We also prove the existence of solutions to the Chern-Simons-Higgs equation $$\Delta u=\lambda e^{u}(e^{u}-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j}$$ on $G$ when $\lambda$ takes the critical value $\lambda_c$ and this completes the results of An Huang, Yong Lin and Shing-Tung Yau (Commun. Math. Phys. 377, 613-621 (2020)).