  We obtain existence results for the $Q$-curvature equation of order $2k$ on a
closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an
integer. We obtain these results under the assumptions that the Yamabe
invariant of order $2k$ is positive and the Green's function of the
corresponding operator is positive, which are satisfied for instance when the
manifold is Einstein with positive scalar curvature. In the case where $2k+1\le
n\le2k+3$ or $(M,g)$ is locally conformally flat, we assume moreover that the
operator has positive mass. In the case where $n\ge2k+4$ and $(M,g)$ is not
locally conformally flat, the results essentially reduce to the determination
of the sign of a complicated constant depending only on $n$ and $k$.
