GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutionsReportar como inadecuado




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1 EDP - Equations aux dérivées partielles IECL - Institut Élie Cartan de Lorraine

Abstract : In this paper we investigate the existence of solutions to a non-linear elliptic problem involving critical Sobolev exponent for a polyharmomic 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 Riemann-ian manifold and adapting Lions-s concentration-compactness lemma. Unlike Coron-s original argument for a bounded domain in 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 ∈ M does not retract to a point in M \{p} is necessary, as shown for the case of the canonical sphere where chopping out a small ball is not enough.





Autor: Saikat Mazumdar -

Fuente: https://hal.archives-ouvertes.fr/



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