On local compactness in quasilinear elliptic problems

Details On local compactness in quasilinear elliptic problems On local compactness in quasilinear elliptic problems

2008-12-04

arxiv.org/abs/0812.0927v1

Differential and integral equations 20, 1 (2006) 77-92

Mathematics

Analysis of PDEs

Scientific paper

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}, Br\'ezis 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.

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