Existence and asymptotics for solutions of a non-local \boldmath $ Q$ -curvature equation in dimension three

Abstract

We study conformal metrics on $ \R{3}$ , i.e., metrics of the form $ g_u=e^{2u}|dx|^2$ , which have constant $ Q$ -curvature and finite volume. This is equivalent to studying the non-local equation \begin{equation*} (-\Delta)^\frac32 u = 2 e^{3u}\text{ in }\R{3},\quad V:=\int_{\R{3}}e^{3u}dx<\infty, \end{equation*} where $ V$ is the volume of $ g_u$ . Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $ V\le 2\pi^2=|\mathbb{S} ^3|$ . Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.