EXISTENCE OF SOLUTIONS TO A GENERALIZED SELF-DUAL CHERN-SIMONS EQUATION ON FINITE GRAPHS

Let G= (V,E) be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equationon G triangle u=lambda e(u)(eu-1)(5)+ 4 pi Sigma(N)(s=1)delta ps, where lambda >0,(delta)p(s) is the Dirac mass at the vertexps, andp1,p2,...,pN are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value lambda such that when lambda >lambda, the generalized Chern-Simons equation has at least two solutions, when lambda=lambda, the generalized Chern-Simons equation has a solution, and when lambda <lambda, the generalized Chern-Simons equation has no solution.