  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.
