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.

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