# An infinite dimensional Saddle Point Theorem and application

## TL;DR

This paper introduces a new Saddle Point Theorem for functionals with infinite-dimensional positive and negative spaces and applies it to solve a type of Schrödinger equation.

## Contribution

A novel Saddle Point Theorem is developed for strongly indefinite functionals using the τ-topology, with application to semilinear Schrödinger equations.

## Key findings

- A new Saddle Point Theorem is established for strongly indefinite functionals.
- The theorem is applied to prove the existence of solutions for a semilinear Schrödinger equation with an indefinite functional.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$J(u) = \dfrac{1}{2} \langle Lu, u \rangle - \Psi (u)$\end{document}J(u)=12〈Lu,u〉−Ψ(u) defined on a Hilbert space X, where \documentclass[12pt]{minimal}
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\begin{document}$L : X \to X$\end{document}L:X→X is a self-adjoint operator whose negative and positive eigenspaces are both infinite-dimensional.