One of the major difficulties in nonlinear elliptic: problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work [7], Brezis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Lapladan homogeneous eigenvalue problem. In this paper, we give EI 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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