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?**



Failure of Palais-Smale Condition C and the Mini-Max Principle
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](http://link.springer.com/chapter/10.1007/BFb0100779) or his book ""Critical Points at Infinity in Some Variational Problem".


