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

We study the existence of solution to the problem $$(-\Delta)^\frac n2u=Qe^{nu}\quad\text{in }\mathbb{R}^{n},\quad \kappa:=\int_{\mathbb{R}^{n}}Qe^{nu}dx<\infty,$$ where $Q\geq 0$, $\kappa\in (0,\infty)$ and $n\geq 3$. Using ODE techniques Martinazzi for $n=6$ and Huang-Ye for $n=4m+2$ proved the existence of solution to the above problem with $Q\equiv const>0$ and for every $\kappa\in (0,\infty)$. We extend these results in every dimension $n\geq 5$, 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 $4$.