An infinite dimensional Saddle Point Theorem and application.

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 $J\left(u\right)=\frac{1}{2}〈Lu,u〉-\Psi \left(u\right)$ defined on a Hilbert space X, where $L:X\to X$ is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.