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 P_g u = u^{2#-1} where P_g u = (Δ_g)^2 u + α Δ_g u + a_α u is a Paneitz-Branson type operator with constant coefficients α and a_α, u is required to be positive, and 2# = 2n/(n-4) is critical from the Sobolev viewpoint. We define the energy function E_m as the infimum of E (u) = ||u||_{2#}^{2#} 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 α_0 >0 such that for α ≥ α_0, the above equation does not have a solution of energy less than or equal to Λ.