Ngô Quốc Anh

December 8, 2010

A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 12:35

Recently, I have read a paper by H. Hofer published in J. London Math. Soc. [here]. In that paper the author studies the behaviour of a functional in the neighborhood of its critical points given by the mountain-pass theorem.

Let us start with the following notations. Assume $F$ is a real Banach space and that $\Phi \in C^1(F, \mathbb R)$. For real numbers $c$ and $d$, we first define the set of critical points with level set $d$ $\displaystyle Cr(\Phi ,d) = \left\{ {u \in F:\Phi (u) = d,\Phi '(u) = 0} \right\}$.

Then we define $\displaystyle Cr(\Phi ) = \bigcup\limits_{d \in \mathbb{R}} {Cr(\Phi ,d)}$.

Next we define $\displaystyle\begin{gathered} {\Phi ^d} = {\Phi ^{ - 1}}(( - \infty ,d]), \hfill \\ {{\dot \Phi }^d} = {\Phi ^{ - 1}}(( - \infty ,d)), \hfill \\ {\Phi _c} = {\Phi ^{ - 1}}([c, + \infty )), \hfill \\ \Phi _c^d = {\Phi ^d} \cap {\Phi _c}. \hfill \\ \end{gathered}$

We say that $\Phi$ satisfies the Palais-Smale condition if the following holds.

If for some sequence $\{u_n\} \in F$ we have $\Phi(u_n) \to d \in \mathbb R$ and $\Phi'(u_n) \to 0$ in $F^\star$ then $\{u_n\}$ is precompact.

Definition. Let $\Phi \in C^1(F, \mathbb R)$ and assume that $u_0 \in Cr(\Phi,d)$. Then

1. $u_0$ is a local minimum if there exists an open neighbourhood $V$ of $u_0$ such that $\Phi(u) \geqslant \Phi(u_0)$ for all $u \in V$;
2. $u_0$ is of mountain-pass type if for all open neighbourhoods $U$ of $u_0$, ${\dot \Phi }^d \cap U\ne \emptyset$ and ${\dot \Phi }^d \cap U$ is not path connected.

Following is the main result of the Hofer paper.

Theorem. Let $\Phi \in C^1(F, \mathbb R)$ satisfy Palais-Smale condition and assume that $e_0, e_1$ are distinct points in $F$. Define $\displaystyle A = \left\{ {a \in C([0,1],F):a(i) = {e_i},i = 0,1} \right\}$

and $\displaystyle d = \mathop {\inf }\limits_{a \in A} \sup \Phi (a([0,1])), \quad c = \max \left\{ {\Phi ({e_0}),\Phi ({e_1})} \right\}$.

Then if $d> c$ the set $Cr(\Phi, d)$ is non-empty. Moreover there exists at least one critical point $u_0$ in $Cr(\Phi, d)$ which is either a local minimum or of mountain-pass type. If all the critical points in $Cr (\Phi, d)$ are isolated in $F$ the set $Cr (\Phi, d)$ contains a critical point of mountain-pass type.

The proof is based on a topological lemma and on the deformation lemma and we leave it for interested reader.