  We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under
conformal normal coordinates around a point, the expansions of Green's function
of $P_g^6$ with pole at this point are presented. As a starting point of the
study of $P_g^6$, we manage to give some existence results of prescribed
$Q$-curvature problem on Einstein manifolds. One among them is that for $n \geq
10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature
and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a
maximum point of $f$ and $f$ satisfies a vanishing order condition at this
maximum point, then there exists a conformal metric $\tilde g$ of $g$ such that
its $Q$-curvature $Q_{\tilde g}^6$ equals $f$.
