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

In this paper we perform a blow-up and quantization analysis of the fractional Liouvilleequationindimension1.Moreprecisely,givenasequenceuk :R→Rofsolutions to (−)12uk=Kkeuk in R, (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,p(RB) and Kkeuk ⇀∗ K∞eu∞ + N πδaj , or (ii) loc ∗ j=1 uk → −∞ uniformly locally in RB and Kkeuk ⇀ Nj=1 αjδaj 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.