Conformally Euclidean metrics on $\mathbb{R}^n$ with arbitrary total Q-curvature

We study the existence of solution to the problem
\[
(-\Delta)^{n/2} u = Q e^{nu} in \mathbb{R}^n, \kappa := \int_{\mathbb{R}^n} Q e^{nu} dx < \infty,
\]
where $Q \ge 0, \kappa \in (0,\infty)$ and $n \ge 3$. Using ODE techniques Martinazzi for n = 6 and Huang-Ye for n = 4m+2 proved the existence of solution to the above problem with Q = const > 0 for every $\kappa \in (0,\infty)$. We extend these results in every dimension $n \ge 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is non-constant, and under some decay assumptions on Q we can also treat the cases n = 3 and 4.