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 Δ u = - λ e g ( u ) e g ( u ) - 1 2 + 4 π ∑ j = 1 N δ p j on G , where λ is a positive constant; g ( u ) is the inverse function of u = f ( υ ) = 1 + υ - e υ on ( - ∞ , 0 ] ; N is a positive integer; p 1 , p 2 , … , p N are distinct vertices of V and δ p j is the Dirac delta mass at p j . We prove that there is critical value λ c such that the generalized Chern–Simons–Higgs equation has a solution if and only if λ ≥ λ c . We also prove the existence of solutions to the Chern–Simons–Higgs equation Δ u = λ e u ( e u - 1 ) + 4 π ∑ j = 1 N δ p j on G when λ takes the critical value λ c and this completes the results of Huang et al. (Commun Math Phys 377:613-621, 2020).