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) = 1/2 Lu, u - Ψ (u) J ( u ) = 1 2 〈 L u , u 〉 − Ψ ( u ) defined on a Hilbert space X , where L : X → X L : X → X is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.