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

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, … , h_n are positive C³ functions, dV_g is the volume form, and ρ_j∈ R⁺(j∈ I). In previous works Lin-Zhang identified a family of hyper-surfaces Γ _N and proved a priori estimates for ρ= (ρ₁, … , ρ_n) in areas separated by Γ _N. Later Lin-Zhang also calculated the leading term of ρ^k- ρ where ρ∈ Γ ₁ is the limit of ρ^k on Γ ₁ and ρ^k is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if ρ^k 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 ρ^k- ρ for ρ∈ Γ _N, but also reveal new robust relations of coefficient functions at different blowup points.