Some existence results for the Toda system on closed surfaces

Given a compact closed surface Sigma, we consider the generalized Toda system of equations on Sigma: -Delta u(i) = Sigma(2)(j)=1 rho(j)a(ij) (h(j)e(j)(u)/integral(Sigma)h(j)(euj)dV(g) - 1) for i = 1, 2, where rho(1), rho(2) are real parameters and h(1), h(2) are smooth positive functions. Exploiting the variational structure of the problem and using a new min-max scheme we prove existence of solutions for generic values of rho(1) and for rho(2) < 4 pi.