Non-compactness of the prescribed Q-curvature problem in large dimensions

Abstract

Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Qg be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to \({P_{g}(u)=c u^{\frac{N+4}{N-4}},\quad u > 0 \quad {\rm on} \; M}\) where Pg is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.