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

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  • 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.


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