Barycenter technique for the higher order $Q$-curvature equation

Let $k\ge1$ be an integer, and $(M,g)$ be a smooth, closed Riemannian manifold of dimension $2k+1\le n\le 2k+3$, or $(M,g)$ be locally conformally flat of dimension $n\ge 2k+1$. Applying the Bahri-Coron barycenter method, we show the existence of a conformal metric with constant $Q$-curvature of order $2k$, or equivalently, the existence of a positive solution for the $2k$-th order $Q$-curvature equation involving the GJMS operator $P_{g}$. We only assume a natural positivity preserving condition on $P_{g}$ and do not suppose any condition on the sign of the {\emph{mass}} of $P_{g}$. In particular, we obtain existence without using a positive mass theorem.