Attaining the optimal constant for higher‐order Sobolev inequalities on manifolds via asymptotic analysis

Abstract Let be a closed Riemannian manifold of dimension , and an integer such that . We show that there exists such that for all , where and . Here, is the optimal constant for the Euclidean Sobolev inequality for all . This result is proved as a consequence of the pointwise blow‐up analysis for a sequence of positive solutions to polyharmonic critical nonlinear equations of the form in . We obtain a pointwise description of , with explicit dependence in as .