Existence of complete conformal metrics on $\mathbb{R}^n$ with prescribed Q-curvature

Abstract

Given a smooth function $f(x)$ on $\mathbb{R}^n$ which is positive somewhere and satisfies $f(x)=O(|x|^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx|^2$ with finite total Q-curvature such that its Q-curvature equals to $f(x)$.