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

Abstract

We study the existence of solution to the problem

$${\left(-\Delta \right)}^{n∕2}u=Q{e}^{nu}\text{ in }{ℝ}^n,\kappa :={\int }_{{ℝ}^n}Q{e}^{nu}dx<\infty ,$$

where $Q\ge 0$, $\kappa \in \left(0,\infty \right)$ and $n\ge 3$. Using ODE techniques, Martinazzi (for $n=6$) and Huang and Ye (for $n=4m+2$) proved the existence of a solution to the above problem with $Q\equiv \text{ constant}>0$ and for every $\kappa \in \left(0,\infty \right)$. 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 nonconstant, and under some decay assumptions on $Q$ we can also treat the cases $n=3$ and $n=4$.