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: { -Delta u(1) = 2 rho(1) (h(1)e(u1)/integral(Sigma)h(1)e(u1)dV(g) - 1) - rho(2) (h(2)e(u2)/integral(Sigma)h(2)e(u2)dV(g) - 1) -4 pi Sigma(m)(j=1)alpha(1,j)(delta(pj) - 1), -Delta u(2) = 2 rho(2) (h(2)e(u2)/integral(Sigma)h(2)e(u2)dV(g) - 1) - rho(1) (h(1)e(u1)/integral(Sigma)h(1)e(u1)dV(g) - 1) -4 pi Sigma(m)(j=1)alpha(2,j)(delta(pj) - 1), which is motivated by the study of models in non-abelian Chern-Simons theory. Here h(1), h(2) are smooth positive functions, rho(1), rho(2) two positive parameters, p(i), points of the surface and alpha(1,i), alpha(2,j) non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation -Delta u = rho(1) (he(u)/integral(Sigma)he(u)dV(g) - 1) - rho(2) (he(-u)/integral(Sigma)he(-u)dV(g) - 1), which arises in fluid dynamics.