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Mean field equations of Liouville type with singular data: sharper estimates. (English) Zbl 1211.35263

Summary: In this and the subsequent paper, we are interested in the following nonlinear equation:
\[ \Delta_g v+ \rho \bigg(\frac{h^* e^v}{\int_M h^* v\,d\mu(x)}-1\bigg)= 4\pi\sum_{j=1}^N\alpha_j(\delta_{q_i}-1)\quad\text{in }M,\tag{1} \]
where \((M,g)\) is a Riemann surface with its area \(|M|=1\); or
\[ \Delta v+ \rho\frac{h^*e^v}{\int_\Omega h^* e^v\,dx}= 4\pi\sum_{j=1}^N \alpha_j \delta_{q_j}\quad\text{in }\Omega, \tag{2} \]
where \(\Omega\) is a bounded smooth domain in \(\mathbb R^2\). 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 (1) or (2) with \(\rho_k\rightarrow\rho_*\). Among other things, we show that for equation (1),
\[ \rho_k-\rho_*= \sum_{j=1}^\tau d_j \big( \Delta \log h^*(p_j)+\rho_*-N^*-2K(p_j)+o(1)\big)e^{-\frac{\lambda_k}{1+\alpha_j}}, \]
and for equation (2),
\[ \rho_k-\rho_*= \sum_{j=1}^\tau d_j \big(\Delta \log h^*(p_j)+o(1)\big)e^{-\frac{\lambda_k}{1+\alpha_j}}, \]
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 [the authors, “A degree counting formulas for singular Liouville-type equation and its application to multi vortices in electroweak theory” (in preparation)] will complete the proof of computing the degree counting formula.

MSC:

35R09 Integro-partial differential equations
35J60 Nonlinear elliptic equations
45G10 Other nonlinear integral equations
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