Let k1 be a positive integer and let Pg be the GJMS operator P₆ of order 2k on a closed Riemannian manifold (M, g) of dimension n>2k. We investigate the compactness of the set of conformal metrics to g with prescribed constant positive Q-curvature of order 2k- or, equivalently, of the set of positive solutions for the 2k-th order Q-curvature equation. Under a natural positivity-preserving condition on P₆ we establish compactness, for an arbitrary 1 k 0 in M. For an arbitrary 1 k < n2, the expression of Pg is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl's celebrated recursive formulae for Pg to perform a refined blow-up analysis for solutions of the Q-curvature equation and to prove a Weyl vanishing result for Pg. This is the first compactness result for an arbitrary 1 k < n2 and the first successful instance where Juhl's formulae are used to yield compactness. Our result also hints that the threshold dimension for compactness for the 2k-th order Q-curvature equation diverges as k +.
Mazumdar et al. (Wed,) studied this question.