Constant Q-Curvature Metrics Near the Hyperbolic Metric

Let (<em>M</em>, <em>g</em>) be a Poincaré-Einstein manifold with a smooth defining function. We prove that there are infinitely many asymptotically hyperbolic metrics with constant <em>Q</em>-curvature in the conformal class of an asymptotically hyperbolic metric close enough to <em>g</em>. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant <em>Q</em>-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.