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,

• is Lipschitz continuous on bounded subsets of H,

• satisfies the Palais–Smale compactness condition,

• ,

• there exist positive constants r and a, and

• .

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, and a far-off spot v) 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

be Banach space. The assumptions of the theorem are:

• and have a Gateaux derivative are endowed with strong topology and weak* topology respectively.

• with

• .

• satisfies weak Palais–Smale condition.

In this case there is a critical point. Moreover, if we define

then

For a proof, see section 5.5 of Aubin and Ekeland.

References

Further reading