*Published Paper*

**Inserted:** 6 feb 2012

**Last Updated:** 25 feb 2013

**Journal:** Ann. Inst. Henri PoincarĂ© (C)

**Year:** 2013

**Doi:** 10.1016/j.anihpc.2012.12.007

**Abstract:**

We study conformal metrics $g_u=e^{2u}

dx

^2$ on $\mathbb{R}^{2m}$ with constant $Q$-curvature $Q_{g_u}\equiv (2m-1)!$ (notice that $(2m-1)!$ is the $Q$-curvature of $S^{2m}$) and finite volume. When $m=3$ we show that there exists $V^*$ such that for any $V\in [V^*,\infty)$ there is a conformal metric $g_u=e^{2u}

dx

^2$ on $\mathbb{R}^{6}$ with $Q_{g_u}\equiv 5!$ and $vol(g_u)=V$. This is in sharp contrast with the four-dimensional case, treated by C-S. Lin. We also prove that when $m$ is odd and greater than $1$, there is a constant $V_m>vol (S^{2m})$ such that for every $V\in (0,V_m]$ there is a conformal metric $g_u=e^{2u}

dx

^2$ on $\mathbb{R}^{2m}$ with $Q_{g_u}\equiv (2m-1)!$, $vol(g)=V$. This extends a result of A. Chang and W-X. Chen. When $m$ is even we prove a similar result for conformal metrics of negative $Q$-curvature.

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