Lyusternik-Schnirelman theory and eigenvalue problems for monotone potential operators

We treat the eigenvalue problem Ax = λBx, where A and B are odd potential operators, A is strictly monotone, bounded, coercive, and continuously invertible, and B is monotone and compact. A naturally defined iteration operator is employed, together with the Lyusternik-Schnirelman theory, to prove the existence of infinitely many nontrivial eigenfunctions. With the possible exception of the multiplicity assertion the results which we obtain are not new. The method which we use, however, has not been applied before to problems of this type. It exploits both the potential character and the monotonicity of the operators and makes the treatment of the infinite dimensional problem essentially as simple as that of its finite dimensional analog. This simplification results primarily from the compactness properties of the iteration operator.