We consider a viscous thin film surrounded by the outer fluid above and a flat horizontal insulator-coated electrode surface below. Our assumption is that the film is completely wetting and conducting. We discuss the global existence of unique solutions perturbed around positive constant solutions whenever applied voltage is sufficiently small after a finite time, and the asymptotic behavior of the solutions. Conversely, when applied voltage is sufficiently large, we show that the solutions around the constant positive solutions are unstable. Moreover, we find the existence of infinitely many bifurcation branches of solutions around positive constant solutions at certain applied voltage and show the stability of nonconstant steady-state solutions.