Sharp Estimates for Solutions of Multi-Bubbles in Compact Riemann Surfaces
Journal
Communications on Pure and Applied Mathematics
Journal Volume
55
Journal Issue
6
Pages
728-771
Date Issued
2002
Author(s)
Abstract
In this paper, we consider a sequence of multibubble solutions U k of the equation (0.1) Δ0u k+ρk(heuk/∫Mheukdμ -1)=0 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 ρk - 8 mπ = 2/m ∑j=1m h-1 (pk, j) (Δ0log h (pk, j) + 8 mπ - 2K (pk, j)) λ k,j e-λk,j + O(e-λk,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 pk converges to 8mπ from below provided that Δ0log 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], which says that any sequence of minimizers u k is uniformly bounded if ρk < 8π and h satisfies Δ0 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], ©2002 Wiley Periodicals, Inc.
Type
journal article