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
$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$
(1)
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. 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 (
\(\alpha _j=\pi \) 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.
