On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

and Applied Analysis 3 Every λk is a critical level of F subject to S, and it is achieved at some uk ∈ S, that is, F|S ′ uk 0 7 cf. 4, 5 . It follows that λk is an eigenvalue of 1 and uk is the corresponding eigenvector. In general, the sequence {λk}k 1 given by 6 does not exhaust the set of all critical levels of F|S, and thus it might not be the set of all eigenvalues of 1 . An eigenvalue of 1 that allows the characterization 6 is called an eigenvalue of Ljusternik-Schnirelmann type. The model example of the abstract setting presented above is the eigenvalue problem for the Dirichlet p-Laplacian. Indeed, set X W 0 Ω , p > 1, and