Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case

Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we consider equations like where is a Paneitz-Branson type operator with constant coefficients α and a_α, u is required to be positive, and is critical from the Sobolev viewpoint. We define the energy function E_m as the infimum of over the u’s which are solutions of the above equation. We prove that E_m (α ) →+∞ as α →+∞ . In particular, for any Λ > 0, there exists α₀ > 0 such that for α ≥ α₀, the above equation does not have a solution of energy less than or equal to Λ.