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.
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