Mean field equations of Liouville type with singular data: Sharper estimates
Chiun‐Chuan Chen, Chang‐Shou Lin · Discrete and Continuous Dynamical Systems · 2010
In this and the subsequent paper, we are interested in thefollowing 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 Diracmeasure at $q$, and both $h^*$'s are positive smooth functions. Inthis paper, we prove a sharp estimate for a sequence of blowing upsolutions $u_k$ to (0.1) or (0.2) with$\rho_k\rightarrow\rho*. Among other things, we show that forequation (0.1),$\rho_k-\rho_*=\sum_{j=1}^\tau d_j( \Delta \logh^*(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 \logh^*(p_j)+o(1))e^{-\frac{\lambda_k}{1+\alpha_j}},$ (0.4)where $\lambda_k\rightarrow+\infty$ and $d_j$ is a constantdepending on $p_j$, a blow up point of $u_k$. 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.