On local compactness in quasilinear elliptic problems. (English) Zbl 1212.35161

Adriouch, Khalid; El Hamidi, Abdallah

Differ. Integral Equ. 20, No. 1, 77-92 (2007).

Summary: One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work H. Brézis and L. Nirenberg [Commun.Pure Appl.Math.36, 437-477 (1983; Zbl 0541.35029)] 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.

MSC:

35J70	Degenerate elliptic equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

quasilinear elliptic problem; local compactness; critical level; Palais-Smale sequence

Citations:

Zbl 0541.35029