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

✍ Scribed by Chiun-Chuan Chen; Chang-Shou Lin

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 322 KB
• Volume: 55
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3014

✦ Synopsis

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 $$\Delta_0 \log h (p_{k,j}) + 8m\pi -2K (p_{k,j}) > 0$$. 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 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.