GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions

In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold $M$. We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist. To perform this topological argument, we overcome the difficulty of dealing with polyharmonic operators on a Riemannian manifold and adapting Lions's concentration-compactness lemma. Unlike Coron's original argument for a bounded domain in $\mathbb{R}^{n}$, we need to do more than chopping out a small ball from the manifold $M$. Indeed, our topological assumption that a small sphere on $M$ centred at a point $p \in M$ does not retract to a point in $M\backslash \{ p \}$ is necessary, as shown for the case of the canonical sphere where chopping out a small ball is not enough.