We consider the SU(3) singular Toda system on a compact surface (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 hi are smooth positive functions on Σ, ρi∈R+, pm∈Σ and αim>−1. We give both existence and non-existence results under some conditions on the parameters ρi and αim. 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."

%B Journal of Functional Analysis %V 270 %P 3750 - 3807 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022123615004942 %R https://doi.org/10.1016/j.jfa.2015.12.011 %0 Journal Article %J Journal of Mathematical Analysis and Applications %D 2015 %T Existence and multiplicity result for the singular Toda system %A Luca Battaglia %K Existence result %K Liouville-type equations %K Multiplicity result %K PDEs on compact surfaces %K Toda system %XWe consider the Toda system on a compact surface (Σ,g)−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−4π∑j=1Jα1j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−4π∑j=1Jα2j(δpj−1), where hi are smooth positive functions, ρi are positive real parameters, pj are given points on Σ and αij are 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."

%B Journal of Mathematical Analysis and Applications %V 424 %P 49 - 85 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022247X14010191 %R https://doi.org/10.1016/j.jmaa.2014.10.081