*Submitted Paper*

**Inserted:** 23 feb 2014

**Last Updated:** 17 jul 2018

**Year:** 2014

**Abstract:**

We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-\Delta)^m u=
Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$,
particularly when $m>1$. This corresponds to finding conformal metrics
$g_u:=e^{2u}

dx

^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume
$V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the
value $V$ and the asymptotic behavior of $u(x)$ as $

x

\to \infty$ can be
simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need
to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the volume $V$ can
be chosen arbitrarily.