%0 Journal Article %J Rend. Sem. Mat. Univ. Padova %D 2016 %T New existence results for the mean field equation on compact surfaces via degree theory %A Aleks Jevnikar %B Rend. Sem. Mat. Univ. Padova %V 136 %P 11–17 %G eng %R 10.4171/RSMUP/136-2 %0 Journal Article %J Advanced Nonlinear Studies %D 2016 %T A note on a multiplicity result for the mean field equation on compact surfaces %A Aleks Jevnikar %B Advanced Nonlinear Studies %I De Gruyter %V 16 %P 221–229 %G eng %R 10.1515/ans-2015-5009 %0 Journal Article %J Advances in Mathematics %D 2015 %T A general existence result for the Toda system on compact surfaces %A Luca Battaglia %A Aleks Jevnikar %A Andrea Malchiodi %A David Ruiz %K Geometric PDEs %K Min–max schemes %K Variational methods %X

In this paper we consider the following Toda system of equations on a compact surface:−Δu1=2ρ1(h1eu1∫Σh1eu1dVg−1)−ρ2(h2eu2∫Σh2eu2dVg−1)−Δu1=−4π∑j=1mα1,j(δpj−1),−Δu2=2ρ2(h2eu2∫Σh2eu2dVg−1)−ρ1(h1eu1∫Σh1eu1dVg−1)−Δu2=−4π∑j=1mα2,j(δpj−1), which is motivated by the study of models in non-abelian Chern–Simons theory. Here h1,h2 are smooth positive functions, ρ1,ρ2 two positive parameters, pi points of the surface and α1,i,α2,j non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation−Δu=ρ1(heu∫ΣheudVg−1)−ρ2(he−u∫Σhe−udVg−1), which arises in fluid dynamics."

%B Advances in Mathematics %V 285 %P 937 - 979 %G eng %U http://www.sciencedirect.com/science/article/pii/S0001870815003072 %R https://doi.org/10.1016/j.aim.2015.07.036 %0 Journal Article %J Analysis & PDE %D 2015 %T A topological join construction and the Toda system on compact surfaces of arbitrary genus %A Aleks Jevnikar %A Kallel, Sadok %A Andrea Malchiodi %B Analysis & PDE %I Mathematical Sciences Publishers %V 8 %P 1963–2027 %G eng %R 10.2140/apde.2015.8.1963 %0 Thesis %D 2015 %T Variational aspects of Liouville equations and systems %A Aleks Jevnikar %K Toda system %I SISSA %G en %1 34676 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by ajevnika@sissa.it (ajevnika@sissa.it) on 2015-08-08T16:04:38Z No. of bitstreams: 1 tesi phd.pdf: 1034249 bytes, checksum: 6988a3a6220b4d1bbd38c3124f520655 (MD5) %0 Journal Article %J Proceedings of the Royal Society of Edinburgh: Section A Mathematics %D 2013 %T An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime %A Aleks Jevnikar %B Proceedings of the Royal Society of Edinburgh: Section A Mathematics %I Royal Society of Edinburgh Scotland Foundation %V 143 %P 1021–1045 %G eng %R 10.1017/S030821051200042X