#### Vol. 10, No. 3, 2017

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

### Ali Hyder

Vol. 10 (2017), No. 3, 635–652
##### Abstract

We study the existence of solution to the problem

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 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 $n=4$.

##### Keywords
$Q$-curvature, nonlocal equation, conformal geometry
##### Mathematical Subject Classification 2010
Primary: 35G20, 35R11, 53A30