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

Abstract 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 ₀ h (p₊, ₉) + 8m -2K (p₊, ₉) > 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. © 2002 Wiley Periodicals, Inc.