  In this paper we prescribe a fourth order conformal invariant on the standard
$n-$sphere, with $n\geq5$, and study the related fourth order elliptic
equation. We first find some existence results in the perturbative case. After
some blow up analysis we build a homotopy to pass from the perturbative case to
the non-perturbative one under some flatness condition. Finally we state some
existence results under the assumption of symmetry.
