From the abstract: “We demonstrate the fact that the famous Ljusternik-Schnirelmann characterization of some eigenvalues of nonlinear elliptic problems (by a minimax formula) has a global variational character. Indeed, we show that, for some homogeneous quasilinear elliptic eigenvalue problems, there are variational eigenvalues other than those of the Ljusternik-Schnirelmann type. In contrast, these eigenvalues have a local variational character. Such a phenomenon does not occur in typical linear elliptic eigenvalue problems.”
The main result of this paper is as follows:
There exist a positive number and a -function of that is -periodic and even about zero, such that, for any , the problem
has two eigenvalues of the Ljusternik-Schnirelmann type with , and a third eigenvalue characterized by
All eigenfunctions associated with have precisely two zeros in the interval . Here, is a special class of curves defined on the submanifold of the Sobolev space
To ensure the existence of a local variational eigenvalue, a special choice of the potential was motivated by the works of P. A. Binding and B. P. Rynne [J. Differ. Equations 235, No. 1, 199–218 (2007; Zbl 1218.34100); J. Differ. Equations 244, No. 1, 24–39 (2008: Zbl 1136.35061)].