Some existence results for the Toda system on closed surfaces

Given a compact closed surface Σ, we consider the generalized Toda system of equations on Σ: −∆ui = ∑2 j=1 ρjaij ( hje uj R Σ hje uj dVg − 1 ) for i = 1, 2, where ρ1, ρ2 are real parameters and h1, h2 are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of ρ1 and for ρ2 < 4π.