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 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 = 8 m π for some positive integer m ≥ 1. We prove among other things that 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 8 m π from below provided that Δ₀ log h (pk,j) + 8mπ -2K (pk,j) > 0 . It generalizes a previous result, due to Ding, Jost, Li, and Wang [18] and Nolasco and Tarantello [31], hich says that any sequence of minimizers u k is uniformly bounded if ρ k > 8π and h satisfies 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].