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

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