Infinite Dimensional Manifolds and Non-Linear Elliptic Eigenvalue Problems

Abstract

with f(u) = c, prescribed, and u lying in a subspace V of a Sobolev space Wm P(Q2) and satisfying a corresponding null boundary condition of variational type. It is our object in the present paper to prove that, if f and g are even functionals, then there exist an infinite number of distinct eigenfunctions uj with g(uj) = c, prescribed. The proof of this result, Theorem II of ? 3 with improvements in Theorems 4.1 and 4.2 of ? 4, consists of two main parts. The first is the proof of an abstract theorem on the critical points of a C2 function on an infinite dimensional Finsler manifold, giving a lower bound on the number of critical points in terms of the Lusternik-Schnirelman category. The second part consists of the application of the abstract theorem to the function f considered on the manifold obtained by identifying the level surface g(u)= c with respect to the involution u > (-u). Section 1 is devoted to the proof of the abstract theorem, Theorem 1.3,