The mountain pass theorem in terms of tangencies

2021/05/15 by Sĩ Tiệp Đinh, Dinh, Si Tiep, Tiến-Sơn Phạm +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2105.07138

openalex publication_date 2021/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses the Mountain Pass Theorem for locally Lipschitz functions on finite-dimensional vector spaces in terms of tangencies. Namely, let f \colon \mathbb Rⁿ → \mathbb R be a locally Lipschitz function with a mountain pass geometry. Let c := inf_{γ∈ \mathcal A}max_t∈[0,1]f(γ(t)), where A is the set of all continuous paths joining x^* to y^*. We show that either c is a critical value of f or c is a tangency value at infinity of f. This reduces to the Mountain Pass Theorem of Ambrosetti and Rabinowitz in the case where the function f is definable (such as, semi-algebraic) in an o-minimal structure.

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