TY - JOUR

T1 - On Liouville systems at critical parameters, Part 2

T2 - multiple bubbles

AU - Huang, Hsin Yuan

AU - Zhang, Lei

N1 - Publisher Copyright:
© 2021, This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply.

PY - 2022/2

Y1 - 2022/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85118989281&partnerID=8YFLogxK

U2 - 10.1007/s00526-021-02110-6

DO - 10.1007/s00526-021-02110-6

M3 - Article

AN - SCOPUS:85118989281

SN - 0944-2669

VL - 61

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 1

M1 - 3

ER -