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

Abstract

Let ( M , g ) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H 2 ( M ) ↪ L 2 ♯ ( M ) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of ( M , g ) . We also prove that we can take ϵ = 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.