Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g)(M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H2(M)↪L2♯(M)H2(M)↪L2♯(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g)(M,g). We also prove that we can take ϵ=0ϵ=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz–Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.