Mountain Passes and Saddle Points

Variational methods find solutions of equations by considering a solution as a critical point of an appropriately chosen function. Local minima and maxima are well-known types of critical points. We explore methods for finding critical points that are neither local maxima or minima, but instead are mountain passes or saddle points. Criteria for the existence of minima or maxima are well-known, but those for mountain passes or saddle points are less well-known. We give an accessible treatment of some criteria for the existence of such points (including the Mountain Pass Lemma), as well as describe a method that could be used to find such points.