A general existence result for the Toda system on compact surfaces

In this paper we consider the following Toda system of equations on a compact surface:-δu1=2ρ1(h1eu1∫σh1eu1dVg-1)-ρ2(h2eu2∫σh2eu2dVg-1)-δu1=-4π∑j=1mα1,j(δpj-1),-δu2=2ρ2(h2eu2∫σh2eu2dVg-1)-ρ1(h1eu1∫σh1eu1dVg-1)-δu2=-4π∑j=1mα2,j(δpj-1), which is motivated by the study of models in non-abelian Chern-Simons theory. Here h1, h2 are smooth positive functions, ρ1, ρ2 two positive parameters, pi points of the surface and α1,i, α2,j non-negative numbers. We prove a general existence result using variational methods.The same analysis applies to the following mean field equation. -δu=ρ1(heu∫σheudVg-1)-ρ2(he-u∫σhe-udVg-1), which arises in fluid dynamics.