Palais–Smale compactness condition

Question

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?

Answer 1

It is rather natural: you are looking for a critical point of a functional $I \colon X \to \mathbb{R}$, where $X$ is a Banach space (in the most used form). Assume that you can, for some reason, construct a sequence $\{u_n\}_n$ of points in $X$ such that $DI(u_n)$ tends to zero and $I(u_n)$ cannot move off to infinity. The Palais-Smale condition tells you that you can always extract a subsequence $u_{n_j}$ converging to an element $u \in X$.

This is very useful, since a mere continuity assumption like $I \in C^1(X)$ will tell you that $DI(u)=0$, i.e. $u$ is a critical point of $I$. Now your task is to construct such a sequence $\{u_n\}_n$. This can be done in at last two ways: the first is a corollary of the Ekeland variational principle. The second is an argument based on some deformation lemma, like in Morse theory. Actually most minimax theorems provide you with a Palais-Smale sequence, and the existence of a critical point comes from a compactness condition. A useful variant of the (PS) condition was introduced by Giovanna Cerami, and it is called Cerami condition. You can find many details in standard books: Willem, Struwe, and so on.