An infinite dimensional saddle point theorem and application

TL;DR

The paper develops a generalized saddle point framework for strongly indefinite functionals by employing the Kryszewski-Szulkin $\tau$-topology, enabling variational methods when both negative and positive subspaces are infinite-dimensional. It proves a deformation lemma and a generalized minimax principle, then establishes an infinite-dimensional saddle point theorem that guarantees a critical value at a computable minimax level. The abstract result is then applied to a semilinear Schrödinger equation with indefinite potential, showing the existence of a nontrivial weak solution under natural spectral and growth assumptions. This work extends Rabinowitz's classical saddle point theory to infinite-dimensional settings, providing a robust variational tool for strongly indefinite problems. The results have potential to impact the analysis of nonlinear PDEs with indefinite quadratic forms by enabling critical point existence where traditional finite-dimensional approaches fail.

Abstract

By using the $τ$-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 $J(u) = \dfrac{1}{2} \langle Lu, u \rangle - Ψ(u)$ defined on a Hilbert space $X$, where $L : X \to X$ is a self-adjoint operator with negative and positive eigenspace both infinite-dimensional.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Lemma 2.1: Deformation Lemma
• Theorem 2.1: General minimax principle
• proof
• Theorem 3.1: Generealized Saddle Point Theorem
• Remark 3
• Corollary 3.1
• Proposition 3.1
• proof