The mountain pass theorem in terms of tangencies

Abstract

This paper addresses the Mountain Pass Theorem for locally Lipschitz functions on finite-dimensional vector spaces in terms of tangencies. Namely, let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a locally Lipschitz function with a mountain pass geometry. Let $$c:=\underset{\gamma \in \mathcal{A}}{\inf }\underset{t\in [0,1]}{\max }f(\gamma (t)),$$ where $\mathcal{A}$ is the set of all continuous paths joining $x^{\ast }$ to $y^{\ast }.$ 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.