Paneitz-Branson invariants on non Einstein manifolds

Let ( M , g ) be a smooth compact Riemannian manifold of dimension n ≥ 5 . Denote P g n the Paneitz-Branson operator. In this paper, we define the Paneitz-Branson invariants μ , μ 1 and μ 2 . We study when they are attained by a metric and this is equivalent to show the existence of positive solutions (and changing-sign solutions) to the nonlinear Paneitz-Branson equation P g n u = C | u | N - 2 u where C is a certain constant and N is a critical exponent.