Constant Q-curvature metrics near the Hyperbolic metric

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.