Existence results for the higher-order Q-curvature equation

We obtain existence results for the Q-curvature equation of order 2k on a closed Riemannian manifold of dimension \(n\ge 2k+1\), where \(k\ge 1\) is an integer. We obtain these results under the assumptions that the Yamabe invariant of order 2k is positive and the Green’s function of the corresponding operator is positive, which are satisfied in particular when the manifold is Einstein with positive scalar curvature. In the case where \(2k+1\le n\le 2k+3\) or the manifold is locally conformally flat, we assume moreover that the operator has positive mass. In the case where \(n\ge 2k+4\) and the manifold is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on n and k.