Existence and multiplicity result for the singular Toda system

We consider the Toda system on a compact surface (Ï, g)-δu1=2Ï1(h1eu1â«Ïh1eu1dVg-1)-Ï2(h2eu2â«Ïh2eu2dVg-1)-4Ïâj=1Jα1j(δpj-1),-δu2=2Ï2(h2eu2â«Ïh2eu2dVg-1)-Ï1(h1eu1â«Ïh1eu1dVg-1)-4Ïâj=1Jα2j(δpj-1), where hiare smooth positive functions, Ïiare positive real parameters, pjare given points on Ï and αijare numbers greater than -1. We give existence and multiplicity results, using variational and Morse-theoretical methods. It is the first existence result when some of the αij's are allowed to be negative