Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

We consider the SU(3)SU(3) singular Toda system on a compact surface (Σ,g)(Σ,g){−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑m=1Mα1m(δpm−1)−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑m=1Mα2m(δpm−1), where hihi are smooth positive functions on Σ, ρi∈R+ρi∈R+, pm∈Σpm∈Σ and αim>−1αim>−1.We give both existence and non-existence results under some conditions on the parameters ρiρi and αimαim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities.