The nonlocal Liouville-type equation in $\mathbb{R}$ 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 $u_k:\mathbb{R}\to \mathbb{R}$ of solutions to \begin{equation} (-Δ)^\frac{1}{2} u_k =K_ke^{u_k}\quad \text{in }\mathbb{R}, \end{equation} 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 $B=\{a_1,\dots ,a_N\}$ and either (i) $u_k\to u_{\infty }$ in $W_{loc}^{1,p}(\mathbb{R}\setminus B)$ and $K_k{e}^{u_k}\stackrel{\ast }{\rightharpoondown }{K}_{\infty }{e}^{u_{\infty }}+{\sum }_{j=1}^N\pi {\delta }_{a_j}$, or (ii) $u_k\to -\infty$ uniformly locally in $\mathbb{R}\setminus B$ and $K_k{e}^{u_k}\stackrel{\ast }{\rightharpoondown }{\sum }_{j=1}^N{\alpha }_j{\delta }_{a_j}$ with ${\alpha }_j\ge \pi$ for every $j$. This result, resting on the geometric interpretation and analysis 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.

arXiv comment38 pages, 12 figures