Conformal metrics in ${\mathbb R}^{2m}$ with constant $Q$-curvature and arbitrary volume

We study the polyharmonic problem $Δ^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that {\sl for any} $V > 0$, there exist radial solutions of $Δ^m u = -e^u$ such that $\int_{{\mathbb R}^{2m}} e^u dx = V.$ It implies that for $m$ odd, given arbitrary volume $V > 0$, there exist conformal metrics $g$ on ${\mathbb R}^{2m}$ with positive constant $Q$-curvature and vol$(g) =V$. This answers some open questions in Martinazzi's work.

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