The nonlocal Liouville-type equation in $$\mathbb {R}$$ R and conformal immersions of the disk with boundary singularities

Abstract

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence $$u_k :\mathbb {R}\rightarrow \mathbb {R}$$ of solutions to 1 $$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$ with $$K_k$$ bounded in $$L^\infty $$ and $$e^{u_k}$$ bounded in $$L^1$$ uniformly with respect to k, we show that up to extracting a subsequence $$u_k$$ can blow-up at (at most) finitely many points $$B=\{a_1,\ldots , a_N\}$$ and that either (i) $$u_k\rightarrow u_\infty $$ in $$W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)$$ and $$K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}$$ , or (ii) $$u_k\rightarrow -\infty $$ uniformly locally in $$\mathbb {R}{\setminus } B$$ and $$K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}$$ with $$\alpha _j\ge \pi $$ for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Riviere and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brezis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates ( $$\alpha _j=\pi $$ and $$\alpha _j\ge \pi $$ ) which are not known in dimension 2 under the weak assumption that $$(K_k)$$ be bounded in $$L^\infty $$ and is allowed to change sign.