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) [GRAPHICS] = 0 in M, where h is a C-2,C-beta positive function in a compact Riemann surface M, and rho(k) is a constant satisfying lim(k-->+infinity), rho(k) = 8Mpi for some positive integer M greater than or equal to 1. We prove among other things that [GRAPHICS] +8mpi - 2K(p(k,j)))lambda(k,j)e(-lambdak,j) + O(e(-lambdak,j)), where p(k,j) are centers of the bubbles of u(k) and lambda(k,j) are the local maxima of u(k) after adding a constant. This yields a uniform bound of solutions as rho(k) converges to 8mpi from below provided that Delta(0) log h (p(k,j)) + 8mpi - 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 rho(k) < 8π and h satisfies Δ(0) log h(p) + 8π - 2K (p) > 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]. (C) 2002 Wiley Periodicals, Inc.