A critical elliptic problem for polyharmonic operators

Yuxin Ge (1) , Juncheng Wei (2) , Feng Zhou (3)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Department of mathematics, Chinese University of Hong Kong
3 Department of Mathematics, East China Normal University

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existence of solutions in domains perforated with the small holes. Some unstable solutions are obtained at higher level sets by Coron's topological method, provided that the minimizing solution does not exist.

Mots clés: Polyharmonic operators; Critical and non-critical dimensions; Ground state solutions; Topological methods

Domaines: Equations aux dérivées partielles [math.AP]

Yuxin Ge, Juncheng Wei, Feng Zhou. A critical elliptic problem for polyharmonic operators. Journal of Functional Analysis, 2011, 260, pp.2247-2282. ⟨10.1016/j.jfa.2011.01.005⟩. ⟨hal-00798829⟩