An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the onedimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations. 0. Introduction Let E be a real Hilbert space with an inner product 〈. , .〉 and let Φ be a twice continuously differentiable functional. Denote the Frechet derivative and the gradient of Φ at x by Φ′(x) and ∇Φ(x) respectively, where as usual 〈∇Φ(x), y〉 := Φ′(x)y ∀ y ∈ E. Recall that a point x0 ∈ E is said to be critical if Φ′(x0) = 0, or equivalently, if ∇Φ(x0) = 0. The level c ∈ R will be called regular if Φ−1(c) contains no critical points, and critical if ∇Φ(x0) = 0 for some x0 ∈ Φ−1(c). Let a, b, a < b, be two regular levels of Φ. Denote M := Φ−1([a, b]) and consider the restriction of Φ to M . In Morse theory one is interested in the local topological structure of the level sets of Φ|M near a critical point and in the relation between this local structure and the topological structure of the set M . To be more specific, suppose that x0 ∈M is an isolated critical point of Φ. Then one defines a sequence of critical groups of Φ at x0 by setting cq(Φ, x0) := Hq(Φ c ∩ U,Φ ∩ U − {x0}), q = 0, 1, 2, ..., (0.1) where c := Φ(x0), Φ c := {x ∈ E : Φ(x) ≤ c}, Hq is the q-th singular homology group with coefficients in some field F and U is a neighbourhood of x0. Define the Morse index of x0 to be the maximal dimension of a subspace of E on which the quadratic form 〈Φ′′(x0)y, y〉 is negative definite. One shows that if x0 is a nondegenerate critical point, i.e., if Φ′′(x0) : E → E is invertible, then cq(Φ, x0) = F Received by the editors March 20, 1995. 1991 Mathematics Subject Classification. Primary 58E05; Secondary 34C25, 35J65, 35L05, 55N20, 58F05.