Constant Q-Curvature Metrics Near the Hyperbolic Metric

Let (M, g) be a Poincaré-Einstein manifold with a smooth defining function. 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.