An infinite dimensional saddle point theorem and application

TL;DR

This paper introduces a new saddle point theorem for strongly indefinite functionals using the $ au$-topology, and applies it to prove the existence of solutions for a class of indefinite semilinear Schrödinger equations.

Contribution

It develops a novel saddle point theorem for strongly indefinite functionals and demonstrates its application to indefinite Schrödinger equations.

Findings

Established a new saddle point theorem using $ au$-topology.

Proved existence of nontrivial solutions for indefinite semilinear Schrödinger equations.

Extended variational methods to strongly indefinite functionals.

Abstract

By using the $\tau$-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 Schr\"odinger equation where the associated functional is indefinite, that is, the functional is of the form $J(u) = \dfrac{1}{2} \langle Lu, u \rangle - \Psi(u)$ defined on a Hilbert space $X$, where $L : X \to X$ is a self-adjoint operator with negative and positive eigenspace both infinite-dimensional.