*Accepted Paper*

**Inserted:** 23 feb 2014

**Journal:** Calc. Var. Partial Differential Equations

**Year:** 2014

**Doi:** 10.1007/s00526-014-0718-9

**Abstract:**

We study conformal metrics on $\mathbb{R}^3$, 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
$(−\Delta)^{3/2}u=2e^{3u}$ in $\mathbb{R}^3$,
$V:= \int_{\mathbb{R}^3} e^{3u} dx<\infty,$
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.