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 is a real Banach space and that . For real numbers and , we first define the set of critical points with level set
Then we define
Next we define
We say that satisfies the Palais-Smale condition if the following holds.
If for some sequence we have and in then is precompact.
Definition. Let and assume that . Then
- is a local minimum if there exists an open neighbourhood of such that for all ;
- is of mountain-pass type if for all open neighbourhoods of , and is not path connected.
Following is the main result of the Hofer paper.
Theorem. Let satisfy Palais-Smale condition and assume that are distinct points in . Define
Then if the set is non-empty. Moreover there exists at least one critical point in which is either a local minimum or of mountain-pass type. If all the critical points in are isolated in the set 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.