The mountain pass theorem in terms of tangencies

TL;DR

This paper extends the Mountain Pass Theorem for locally Lipschitz functions by characterizing critical and tangency values, especially in the context of definable functions within o-minimal structures.

Contribution

It introduces a new perspective on the Mountain Pass Theorem using tangencies, applicable to a broader class of functions including semi-algebraic ones.

Findings

Either the mountain pass level is a critical value or a tangency value at infinity.

Reduces to classical Mountain Pass Theorem for definable functions.

Provides a new framework for analyzing variational problems with tangency considerations.

Abstract

This paper addresses the Mountain Pass Theorem for locally Lipschitz functions on finite-dimensional vector spaces in terms of tangencies. Namely, let $f \colon \mathbb R^n \to \mathbb R$ be a locally Lipschitz function with a mountain pass geometry. Let $$c := \inf_{\gamma \in \mathcal A}\max_{t\in[0,1]}f(\gamma(t)),$$ where $\mathcal{A}$ 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.