## Diff. Geom, Math. Phys., PDE Seminar: Saikat Mazumdar

- Date: 10/18/2016
- Time: 15:30

*UBC*

University of British Columbia

Higher order elliptic problems with critical sobolev growth on a compact riemannian manifold: Best constants and existence

We investigate the existence of solutions to a nonlinear elliptic problem involving the 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 and the manifold satisfies a certain topological assumption. To perform the topological argument, we obtain a decomposition of Palais-Smale sequences as a sum of bubbles and adapt Lions's concentration-compactness lemma.

Location: ESB 2012