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 generalized Chern–Simons–Higgs equation $$\begin{aligned} \Delta u=-\lambda e^{g(u)}\left( e^{g(u)}-1\right) ^2+4\pi \sum \limits _{j=1}^{N}\delta _{p_j} \end{aligned}$$on G, where \(\lambda \) is positive constant; g(u) inverse function \(u=f(\upsilon )=1+\upsilon -e^{\upsilon }\) on \((-\infty , 0]\); N integer; \(p_1, p_2, \ldots p_N\) are distinct vertices V and \(\delta _{p_j}\) Dirac delta mass at \(p_j\). prove that there critical value _c\) such has solution if only \ge \lambda . also u=\lambda e^{u}(e^{u}-1)+4\pi G when takes this completes results Huang et al. (Commun Math Phys 377:613-621, 2020).