COMPACTNESS AND GLOBAL ESTIMATES FOR A FOURTH ORDER EQUATION OF CRITICAL SOBOLEV GROWTH ARISING FROM CONFORMAL GEOMETRY

Abstract

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like P g u = u 2 ♯ -1 , where [Formula: see text] is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and [Formula: see text] is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.