## T. Jin - A. Maalaoui - L. Martinazzi - J. Xiong

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

created by martinazz on 17 Jul 2018

[

BibTeX]

*preprint*

**Inserted:** 17 jul 2018

**Year:** 2013

**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 $$ (-\Delta)^{\frac32} u = 2
e^{{3u}$$} in $R^3$ $$V:=\int_{{\mathbb{R}}^{3}e}^{{3u}dx<\infty,$$} 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=

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.