  We prove that certain subcritical Paneitz-Branson type equations on a closed
Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positIve solutions,
where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This
implies that if $(X,h)$ is a closed positive Einstein manifold then for $\ep
>0$ small enough there are at least $\mathrm{Cat}(M)$ metrics of constant
$Q$-curvature in the conformal class of the Riemannian product $g+\ep h$.
