TITLE: Failure of Palais-Smale Condition C and the Mini-Max Principle
QUESTION [5 upvotes]: To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition C of Palais and Smale (note that this is automatically satisfied for $M$ compact). It ultimately implies (for instance) the Deformation Lemma and the Mini-Max Principle.
But what if Condition C fails? I'm interested in to what extent I lose control over the function. For instance, I am aware of Uhlenbeck's "perturbation method", where if $f$ doesn't satisfy Condition C then we can look at $f_\varepsilon=f+\varepsilon\cdot g$ (fix function $g$) that satisfies Condition C and try to get critical points of $f$ as limits of those of $f_\varepsilon$.
When Condition C fails, is there a work-around to save the Mini-Max Principle?

REPLY [2 votes]: There are some works on this, but maybe THE expert on this topic is Abbas Bahri. Take a look at his works or google "critical point at infinity". 
See, for example: this paper or his book ""Critical Points at Infinity in Some Variational Problem".