Blow-up analysis of a nonlocal Liouville-type equation

Abstract

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 $(-\Delta {)}^{\frac{1}{2}}$ stands for the fractional Laplacian and $\kappa$ 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 $\mathbb{R}$.

59 pages

References in corpus (1)

• Concentration-compactness phenomena in the higher order Liouville's equation

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