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

We study the polyharmonic problem $$\Delta ^m u = \pm e^u$$ in $$\mathbb {R}^{2m}$$ , with $$m \ge 2$$ . In particular, we prove that for any $$V > 0$$ , there exist radial solutions of $$\Delta ^m u = -e^u$$ such that $$\begin{aligned} \int _{\mathbb {R}^{2m}} e^u dx = V. \end{aligned}$$ It implies that for m odd, given any $$Q_0 >0$$ and arbitrary volume $$V > 0$$ , there exist conformal metrics g on $$\mathbb {R}^{2m}$$ with constant Q-curvature equal to $$Q_0$$ and vol $$(g) =V$$ . This answers some open questions in Martinazzi’s work (Ann IHP Analyse non lineaire 30:969–982, 2013).