Mean field equations of Liouville type with singular data: Sharper estimates
Journal
Discrete and Continuous Dynamical Systems
Journal Volume
28
Journal Issue
3
Pages
1237-1272
Date Issued
2010
Author(s)
Abstract
In this and the subsequent paper, we are interested in the following nonlinear equation: where (M, g) is a Riemann surface with its area |M| = 1; or (equation) where Ω is a bounded smooth domain in ℝ2. Here, ρ, αj are positive constants, δq is the Dirac measure at q, and both hz.ast;'s are positive smooth functions. In this paper, we prove a sharp estimate for a sequence of blowing up solutions uk to (0.1) or (0.2) with ρk → ρ* Among other things, we show that for equation (0.1), (equation) and for equation (0.2), (equation) where λk → +∞ and dj is a constant depending on pj, a blow up point of uk See section 1 for more precise description. These estimates play an important role when the degree counting formulas are derived. The subsequent paper [19] will complete the proof of computing the degree counting formula.
Type
journal article