Mean field equations of Liouville type with singular data: Sharper estimates

In this and the subsequent paper, we are interested in thefollowing nonlinear equation: g v+ (h^* eᵛM h^* vd (x) -1) =4₉=₁Nⱼ (ₐ㶁-1) M, (0. 1) where (M, g) is a Riemann surface with its area |M|=1; or v+^*eᵛ_ h^*eᵛdx=4₉=₁Nⱼ ₐ䲛, (0. 2) where is a bounded smooth domain in R². Here, , ⱼ are positive constants, q is the Diracmeasure at q, and both h^*'s are positive smooth functions. Inthis paper, we prove a sharp estimate for a sequence of blowing upsolutions uₖ to (0. 1) or (0. 2) withₖ*. Among other things, we show that forequation (0. 1), ₖ-_*=₉=₁^ dⱼ (^* (pⱼ) +_*-N^*-2K (pⱼ) +o (1) ) e^-ₖ{1+ⱼ}, (0. 3) and for equation (0. 2), ₖ-_*=₉=₁^ dⱼ (^* (pⱼ) +o (1) ) e^-ₖ{1+ⱼ}, (0. 4) where ₖ+ and dⱼ is a constantdepending on pⱼ, a blow up point of uₖ. See section 1 formore precise description. These estimates play an important rolewhen the degree counting formulas are derived. The subsequentpaper 19 will complete the proof of computing thedegree counting formula.