Lusternik-Schnirelman category and nonlinear elliptic eigenvalue problems
Felix E. Browder · Bulletin of the American Mathematical Society · 1965
Introduction.Let £2 be a bounded, smoothly bounded open subset of R n (or of a differentiate manifold), ƒ and g two real-valued functional of the formdefined for r-vector functions u on Q.Let A and B be the Euler-Lagrange systems for ƒ and g respectively, i.e.(In a preceding note [3], we observed that under assumptions of polynomial growth on F, G, F Pa , and G Pa in u and its derivatives, ellipticity and positivity for B, and positivity for A, there exists an eigenf unction of the pair (A, B), i.e. a solution u of the equation Bu=\Au with X in R 1 , with f(u) prescribed and u satisfying a null variational boundary condition corresponding to a given closed subspace V of a Sobolev space TF m » p (Q).It is our object in the present note to summarize the principal results of the writer's paper [5], where it is shown that if in addition ƒ and g are even functionals of u % then there exist an infinite number of distinct eigenf unctions Uj with g{u J )=c 1 prescribed.This result is based in turn upon the estimation from below of the number of critical points of a real-valued function ƒ on an infinite dimensional Finsler manifold M in terms of the Lusternik-Schnirelman category of M.1. Let M be an infinite-dimensional manifold of class C 2 modelled on the reflexive Banach space B (cf. [7], T(M) and T*(M) the tangent and cotangent spaces of M, respectively.A Finsler structure on M is a function p: T*(M)-KR 1 such that p