PROBLÈMES ELLIPTIQUES D’ORDRE SUPÉRIEUR SUR LES VARIÉTÉS RIEMANNIENNES

Abstract

The aim of this thesis is to study two problems arising from conformal geometry. Firstly, we are interested in the existence of nodal solutions for fourth order elliptic equations involving the Paneitz-Branson type operator on a compact Riemannian manifold with boundary of dimension n > 5. For the second one, we are interested in the existence of nodal solutions for elliptic equations involving the GJMS type operator on a compact Riemannian manifold with boundary of dimension n > 2k where k ∈ N ? . The two problems have the peculiarity of containing the critical Sobolev exponent, which leads us to use the variational approach developed by H. Yamabe