On Liouville systems at critical parameters, Part 2: multiple bubbles

Hsin Yuan Huang, Lei Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper, we continue to consider the generalized Liouville system: Δgui+∑j=1naijρj(hjeuj∫MhjeujdVg-1)=0inM,i∈I={1,…,n},where (M, g) is a Riemann surface M with volume 1, h1, … , hn are positive C3 functions, dVg is the volume form, and ρj∈ R+(j∈ I). In previous works Lin-Zhang identified a family of hyper-surfaces Γ N and proved a priori estimates for ρ= (ρ1, … , ρn) in areas separated by Γ N. Later Lin-Zhang also calculated the leading term of ρk- ρ where ρ∈ Γ 1 is the limit of ρk on Γ 1 and ρk is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if ρk tends to a higher order hypersurface Γ N (N> 1). Over the years numerous attempts have failed but in this article we overcome all the stumbling blocks and completely solve the problem under the most general context: We not only capture the leading terms of ρk- ρ for ρ∈ Γ N, but also reveal new robust relations of coefficient functions at different blowup points.

Original languageEnglish
Article number3
JournalCalculus of Variations and Partial Differential Equations
Issue number1
StatePublished - Feb 2022


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