# American Institute of Mathematical Sciences

July  2010, 28(3): 1237-1272. doi: 10.3934/dcds.2010.28.1237

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

 1 Department of Mathematics, National Center for Theoretical Sciences, National Taiwan University, Taipei, 106, Taiwan, Taiwan

Received  April 2010 Published  April 2010

In this and the subsequent paper, we are interested in the following nonlinear equation:

$\Delta_g v+\rho(\frac{h^* e^v}{\int_M h^* vd\mu(x)}-1)= 4\pi\sum_{j=1}^N\alpha_j(\delta_{q_i}-1)\quad\text{in }M,$(0.1)

where $(M,g)$ is a Riemann surface with its area $|M|=1$; or

$\Delta v+\rho\frac{h^*e^v}{\int_\Omega h^* e^vdx}=4\pi\sum_{j=1}^N\alpha_j \delta_{q_j}\quad\text{in }\Omega,$ (0.2)

where $\Omega$ is a bounded smooth domain in $R^2$. Here, $\rho, \alpha_j$ are positive constants, $\delta_q$ is the Dirac measure at $q$, and both $h^*$'s are positive smooth functions. In this paper, we prove a sharp estimate for a sequence of blowing up solutions $u_k$ to (0.1) or (0.2) with $\rho_k\rightarrow\rho*. Among other things, we show that for equation (0.1),$\rho_k-\rho_*=\sum_{j=1}^\tau d_j( \Delta \log h^*(p_j)+\rho_*-N^*-2K(p_j)+o(1) )e^{-\frac{\lambda_k}{1+\alpha_j}}, $(0.3) and for equation (0.2),$ \rho_k-\rho_*=\sum_{j=1}^\tau d_j(\Delta \log h^*(p_j)+o(1))e^{-\frac{\lambda_k}{1+\alpha_j}},$(0.4) where$\lambda_k\rightarrow+\infty$and$d_j$is a constant depending on$p_j$, a blow up point of$u_k$. 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. Citation: Chiun-Chuan Chen, Chang-Shou Lin. 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