*Published Paper*

**Inserted:** 6 feb 2009

**Last Updated:** 17 jul 2018

**Journal:** Rend. Lincei Mat. Appl.

**Volume:** 19

**Pages:** 279-292

**Year:** 2008

**Abstract:**

We study the conformal metrics on $\R^{2m}$ with constant Q-curvature $Q$
having finite volume, particularly in the case $Q\leq 0$. We show that when
$Q<0$ such metrics exist in $\R^{2m}$ if and only if $m>1$. Moreover we study
their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we
treated in a recent paper. When Q=0, we show that such metrics have the form
$e^{2p}g_{\R^{2m}}$, where $p$ is a polynomial such that $2\leq \deg p\leq
2m-2$ and $\sup_{\R^{2m}}p<+\infty$. In dimension 4, such metrics are exactly
the polynomials $p$ of degree 2 with $\lim_{

x

\to+\infty}p(x)=-\infty$.

**Keywords:**
Q-curvature, Concentration-compactness, Conformal geometry

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