  Let $(M,\,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth
defining function. In this note, we prove that there are infinitely many
asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal
class of an asymptotically hyperbolic metric close enough to $g$. These metrics
are parametrized by the elements in the kernel of the linearized operator of
the prescribed constant $Q$-curvature equation. A similar analysis is applied
to a class of fourth order equations arising in spectral theory.
