Abstract 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) = 12 Lu, u - (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.
Colin et al. (Wed,) studied this question.