The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points. Statement The assumptions of the theorem are: is a functional from a Hilbert space H to the reals, and is Lipschitz continuous on bounded subsets of H, satisfies the Palais–Smale compactness condition, , there exist positive constants r and a such that if , and there exists with such that . If we define: and: then the conclusion of the theorem is that c is a critical value of I. Visualization The intuition behind the theorem is in the name "mountain pass." Consider I as describing elevation. Then we know two low spots in the landscape: the origin because , and a far-off spot v where . In between the two lies a range of mountains (at ) where the elevation is high (higher than a>0). In order to travel along a path g from the origin to v, we must pass over the mountains—that is, we must go up and then down. Since I is somewhat smooth, there must be a critical point somewhere in between. (Think along the lines of the mean-value theorem.) The mountain pass lies along the path that passes at the lowest elevation through the mountains. Note that this mountain pass is almost always a saddle point. For a proof, see section 8.5 of Evans. Weaker formulation Let be Banach space. The assumptions of the theorem are: and have a Gateaux derivative which is continuous when and are endowed with strong topology and weak* topology respectively. There exists such that one can find certain with . satisfies weak Palais–Smale condition on . In this case there is a critical point of satisfying . Moreover, if we define then For a proof, see section 5.5 of Aubin and Ekeland. References Further reading Mathematical analysis Calculus of variations Theorems in analysis