On local compactness in quasilinear elliptic problems

Khalid Adriouch (1) , Abdallah El Hamidi (1)

1 LMA-Rochelle - Laboratoire de Mathématiques et Applications

One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Brézis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact.

Mots clés: local compactness; critical exponent; best sobolev constant

Domaines: Equations aux dérivées partielles [math.AP]

Khalid Adriouch, Abdallah El Hamidi. On local compactness in quasilinear elliptic problems. Differential and integral equations, 2006, 20 (1), pp.77-92. ⟨hal-00344212⟩