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

TitleExistence and non-existence results for the SU(3) singular Toda system on compact surfaces
Publication TypeJournal Article
Year of Publication2016
AuthorsBattaglia, L, Malchiodi, A
JournalJournal of Functional Analysis
Volume270
Pagination3750 - 3807
ISSN0022-1236
KeywordsLiouville-type equations; Min–max solutions; Non-existence results; Toda system
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 Pohožaev-type identities."

URLhttp://www.sciencedirect.com/science/article/pii/S0022123615004942
DOI10.1016/j.jfa.2015.12.011

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