## Finite dimensional “Mountain Pass Lemma”

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Question Does anyone know of a good reference which I can cite for the finite dimensional version of Mountain Pass Lemma?

Motivation I am writing a paper and found myself using the following result:

Let $f$ be a proper smooth real-valued function on $\mathbf{R}^3$ such that $f(0) = 0$, $f|_{B_1(0)} \geq 0$, $f|_{\partial B_1(0)} \geq 1$ and $\exists p \in {\partial B_2(0)}$ such that $f(p) = 0$. Then $\exists q\in \mathbf{R}^3 \setminus B_1(0)$ such that $f'(q) = 0$ and $f(q) \geq 1$.

For the time being I referred to Ambrosetti and Rabinowitz's JFA article for the mountain pass lemma, but citing a Banach space version for a finite-dimensional Euclidean space application gives me a funny feeling. (Also, if feels like such a result could in principle be found in not-so-advanced undergraduate textbooks...)

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1
I colleague in grad school some years ago had exactly the same difficulty. Alas, he also used Ambrosetti and Rabinowitz. – Marc Chamberland Aug 30 at 15:14

2
Only somewhat related: Mike Usher has an interesting article about a converse of this (in finite dimensions) arxiv.org/abs/1207.0889 – Sam Lisi Aug 31 at 7:07

## 4 Answers

My book An Invitation to Morse Theory, 2nd Edition, Springer Verlag 2011 describes the finite dimensional Mountain Pass Lemma in Example 2.53. There I work on a compact manifold, but the compactness of the manifold can be substituted by a properness assumption on the function. In the same section I explain a more general Min-Max principle (Thm. 2.51) and in Example 2.53 I explain how this implies the Mountain Pass Lemma.

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@Liviu: a small technical question. In your book you assume that $f$ is $C^\infty$, do you have any idea about lower regularity? The classical MPT is of course proven for $C^1$. But in your proof of the deformation lemma you use ODE existence to construct the flow, which usually uses Picard (I'm not sure if Peano would be able to give you that the flow generates a homeo/diffeomorphism) and requires $C^2$ or $C^{1,1}$ in your function.  – Willie Wong Aug 31 at 14:20
As you correctly pointed out $C^{1,1}$ suffices. All one needs is that the gradient vector field be locally Lipschitz to invoke  existence results. – Liviu Nicolaescu Aug 31 at 14:32

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For historical interest: A friend pointed me to the book

• Youssef Jabri, The Mountain Pass Theorem: Variants, Generalizations and Some Applications, CUP

which asserts that one of the earliest known published version of the finite dimensional mountain pass theorem was due to

• Richard Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience

published originally in 1950. The version stated and proven by Courant does not, technically speaking, imply the result I stated in the question text (the points $0$ and $p$ are assumed to be local minima of the function $f$). But a simple modification of the deformation lemma (for example, as in Liviu's book that he mentioned) would do.

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This seems to be in L Evans's PDE book, section 8.5

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I have stumbled across Richard Palais' (co-author Chuu-lian Terng) Critical Point Theory and Submanifold Geometry (Springer Lecture Notes in Math 1353). This is an awesome book!

The "Mountain Pass Lemma" for finite-dimensional manifolds is presented as Theorem 9.2.7 (pg189).

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