Mountain pass theorem

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.