Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 35J60; Secondary: 45G10.

 Citation: