Existence results for the higher-order $Q$-curvature equation

Abstract

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 for instance when the manifold is Einstein with positive scalar curvature. In the case where $2k+1\le n\le 2k+3$ or $(M,g)$ is locally conformally flat, we assume moreover that the operator has positive mass. In the case where $n\ge 2k+4$ and $(M,g)$ is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on $n$ and $k$.

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