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A general existence result for the Toda system on compact surfaces

TitleA general existence result for the Toda system on compact surfaces
Publication TypeJournal Article
Year of Publication2015
AuthorsBattaglia, L, Jevnikar, A, Malchiodi, A, Ruiz, D
JournalAdvances in Mathematics
Volume285
Pagination937 - 979
ISSN0001-8708
KeywordsGeometric PDEs; Min–max schemes; Variational methods
Abstract

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."

URLhttp://www.sciencedirect.com/science/article/pii/S0001870815003072
DOI10.1016/j.aim.2015.07.036

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