Lusternik-Schnirelman theory and non-linear eigenvalue problems

Abstract

In this paper we employ the ideas of Lusternik and Schnirelman [14] to establish the existence of infinitely many distinct eigenfunctions for problem (1.1). This problem has already attracted considerable interest [3-7, 9-11, 13, 14, 18, 19]. All these investigations have been based on variants of the so-called Lusternik-Schnirelman theory. In the early 1930's Lusternik and Schnirelman developed a theory of critical points for differentiable functions on finite-dimensional Riemannian manifolds. One of the principal tools for establishing the existence of "intermediate" critical points (i.e. of critical points not belonging to absolute maxima or minima) is the same as in the Morse theory, namely the deformation of the manifold along gradient lines. The application of this theory to infinite-dimensional eigenvalue problems of the form (1.1) which arise in connection with differential and integral equations require the generalization of the Lusternik-Schnirelman theory to infinite-dimensional manifolds. This extension has been made by Schwartz [16, 17] for Riemannian manifolds modelled on Hilbert spaces and by R. S. Palais [15] for Finsler manifolds modelled on arbitrary Banach spaces. These generalizations are based on a fundamental compactness assumption, the so-called Palais-Smale Condition. In applying this general Lusternik-Schnirelman theory to the eigenvalue problem (1.1) one is faced with two technical difficulties. First, one