An infinite dimensional saddle point theorem and application

Abstract

By using the $\ddot{I}„$-topology of Kryszewski and Szulkin, we establish a natural new version of the Saddle Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of a nontrivial solution of the strongly indefinite semilinear Schrödinger equation where the associated functional is indefinite, that is, the functional is of the form $J(u)=\frac{1}{2}⟨Lu,u⟩-\hat{I}¨(u)$ defined on a Hilbert space $X$, where $L:X\to X$ is a self-adjoint operator with negative and positive eigenspace both infinite-dimensional.