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)) (e(g(u)) - 1)(2) + 4 pi Sigma(N)(j=1) (delta)p(j) on G, where lambda is a positive constant; g(u) is the inverse function of u = f (nu) = 1+nu - e(nu) on (-infinity, 0]; N is a positive integer; p(1), p(2),..., p(N) are distinct vertices of V and delta(pj) 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 >= 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 Sigma(N delta)(j=1)pj on G when lambda takes the critical value.c and this completes the results of Huang et al. (Commun Math Phys 377:613-621, 2020).