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

In this paper, we consider a sequence of multibubble solutions uk of the equation 0.1 $$\Delta_0 u_k + \rho_k \left( {{h e^{u_k}}\over{\int_M h e^{u_k} \,d\mu}}-1\right) = 0 \quad \hbox{in} \ M\,,$$ where h is a C2,β positive function in a compact Riemann surface M, and ρk is a constant satisfying limk→+∞ ρk = 8mπ for some positive integer m ≥ 1. We prove among other things that $$\rho_k - 8m\pi ={{2}\over{m}} \sum_{j=1}^m h^{-1} (p_{k,j}) (\Delta_0 \log h (p_{k,j})$$$$+8m\pi -2K (p_{k,j})) \lambda_{k,j} e^{-\lambda_{k,j}} +O(e^{-\lambda_{k,j}})\,,$$ where pk,j are centers of the bubbles of uk and λk,j are the local maxima of uk after adding a constant. This yields a uniform bound of solutions as ρk converges to 8mπ from below provided that $$\Delta_0 \log h (p_{k,j}) + 8m\pi -2K (p_{k,j}) > 0$$. 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 uk is uniformly bounded if ρk > 8π and h satisfies $$\Delta_0 \log h (p) + 8 \pi -2K (p) > 0$$ 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]. © 2002 Wiley Periodicals, Inc.