An infinite dimensional Saddle Point Theorem and application

By using the τ-topology of Kryszewski and Szulkin, we establish a natural new version of the Saddle Point Theorem for strongly indefinite functionals. The abstract result will be applied to study the existence of solutions to a strongly indefinite semilinear Schrödinger equation, where the associated functional is indefinite, that is, the functional is of the form J(u) = ½〈Lu, u〉 - Ψ(u) defined on a Hilbert space X, where L:X → X is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.