Blow-up analysis of a nonlocal Liouville-type equation

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In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation equation(-Δ)12 u= κeᵘ-1 in S¹, equation where (-Δ)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 R equation (-Δ)¹/2 u =Keᵘ in R, equation with K bounded on R.