一类非局部Liouville型方程的爆破分析
Blow-up analysis of a nonlocal Liouville-type equation
摘要：
本文对非局部Liouville型方程\begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation}进行了爆破和量子化分析，其中$(-Δ)^\frac{1}{2}$表示分数拉普拉斯算子，$κ$是有界函数.我们将上述方程解释为曲线在保角参数化下的曲率方程。我们还建立了这个方程与$\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation}中的类似方程之间的关系，其中$K$有界于$\mathbb{R}$ .
英文摘要：
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-Δ)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.
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