The nonlocal Liouville-type equation in R and conformal immersions of the disk

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence uk:R→R of solutions to (−Δ)12uk=KkeukinR,(1) with Kk bounded in L∞ and euk bounded in L1 uniformly with respect to k, we show that up to extracting a subsequence uk can blow-up at (at most) finitely many points B={a1,...,aN} and that either (i) uk→u∞ in W1,ploc(R∖B) and Kkeuk⇀∗K∞eu∞+∑Nj=1πδaj, or (ii) uk→−∞ uniformly locally in R∖B and Kkeuk⇀∗∑Nj=1αjδaj with αj≥π for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis-Merle and Li-Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (αj=π and αj≥π) which are not known in dimension 2 under the weak assumption that (Kk) be bounded in L∞ and is allowed to change sign.