# Blow-up analysis of a nonlocal Liouville-type equation

Martinazzi, Luca and Riviere, Tristan and Da Lio, Francesca. (2015) Blow-up analysis of a nonlocal Liouville-type equation.

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Official URL: https://edoc.unibas.ch/69986/

$(1) (-\Delta)^{1/2} u = \kappa e^u - 1 in S^1$
where $(-\Delta)^{1/2}$ stands for the fractional Laplacian and $\kappa$ is a bounded function. The equation (1) can actually be interpreted as the prescribed curvature equation to a curve in conformal parametrization. Thanks to this geometric interpretation we perform a subtle blow-up and quantization analysis of (1). We also show a relation between equation (1) and the analogous equation in $\mathbb{R}$
$(2) (-\Delta)^{1/2} u = K e^u in \mathbb{R},$
with K bounded on $\mathbb{R}$.