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 $f \colon \mathbb R^n \to \mathbb R$ 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.