Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces

In this paper, we consider a sequence of multibubble solutions u k of the equation (0.1) Δ 0 u k + ρ k (he uk /f M he u k d μ -1)=0 in M, where h is a C 2,β positive function in a compact Riemann surface M, and ρ k is a constant satisfying lim k→+ ∞ ρ k = 8mπ for some positive integer m ≥ 1. We prove among other things that ρ k - 8mπ = 2/m m Σ/j=1h -1 (p k,j )(Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ))λ k,j e -λk,j + O(e -λk,j ), where p k,j are centers of the bubbles of u k and λ k,j are the local maxima of u k after adding a constant. This yields a uniform bound of solutions as ρ k converges to 8mπ from below provided that Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ) > 0. It generalizes a previous result, due to Ding, Jost, Li, and Wang [18] and Nolasco and Tarantello [31], which says that any sequence of minimizers u k is uniformly bounded if ρ k 0 for any maximum point p of the sum of 2 log h and the regular part of the Green function, where K is the Gaussian curvature of M. The analytic work of this paper is the first step toward computing the topological degree of (0.1), which was initiated by Li [24].