This resembles the situation of a traveler trying to cross a mountain range without climbing higher than necessary. If we can find a continuous path connecting the two points which does not take the traveler higher than any other such path, it is expected that this path will produce a critical point.

However, there is a difficulty which must be addressed. One must allow the competing path to roam freely, and conceivably they can take the traveler to infinity while he is trying to cross some local mountains.

We will see in this chapter the so-called bounded MPT of Schechter and the mountain impasse theorem of Tintarev. They correspond to the situation where the functional does not necessarily satisfy the Palais-Smale condition but still has the geometry of the MPT. The peculiarity of these two results is that they both require the continuous paths appearing in the minimaxing procedure of the MPT to be within a level set of some auxiliary function. This adds enough compactness to give some interesting results.