This paper is concerned with nonlinear eigenvalue problems
for which not all eigenvalues are of variational type in the sense of Ljusternik-Schnirelman (1934; Zbl 0011.02803); see e.g. H. Amann [Math. Ann. 199, 55–71 (1972; Zbl 0233.47049)]. Here is the -Laplacian with , is a smooth domain in , and are coefficients with on .
Some examples are given for ordinary differential equations with periodic boundary conditions and partial differential equations with Neumann boundary conditions, in the case of non-constant coefficients. Moreover, for the periodic problem, the variational eigenvalues are characterized via an extremal property within the set of Carathéodory eigenvalues.