@article {BATTAGLIA20163750, title = {Existence and non-existence results for the SU(3) singular Toda system on compact surfaces}, journal = {Journal of Functional Analysis}, volume = {270}, number = {10}, year = {2016}, pages = {3750 - 3807}, abstract = {

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 Poho{\v z}aev-type identities."

}, keywords = {Liouville-type equations, Min{\textendash}max solutions, Non-existence results, Toda system}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2015.12.011}, url = {http://www.sciencedirect.com/science/article/pii/S0022123615004942}, author = {Luca Battaglia and Andrea Malchiodi} } @article {BATTAGLIA201549, title = {Existence and multiplicity result for the singular Toda system}, journal = {Journal of Mathematical Analysis and Applications}, volume = {424}, number = {1}, year = {2015}, pages = {49 - 85}, abstract = {

We 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{\textquoteright}s are allowed to be negative."

}, keywords = {Existence result, Liouville-type equations, Multiplicity result, PDEs on compact surfaces, Toda system}, issn = {0022-247X}, doi = {https://doi.org/10.1016/j.jmaa.2014.10.081}, url = {http://www.sciencedirect.com/science/article/pii/S0022247X14010191}, author = {Luca Battaglia} } @mastersthesis {2015, title = {Variational aspects of Liouville equations and systems}, year = {2015}, note = {The PHD thesis is composed of 112 pages and is recorded in PDF format}, school = {SISSA}, keywords = {Toda system}, author = {Aleks Jevnikar} }