Mountain pass theorem

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.

[1][2] Given certain conditions on a function, the theorem demonstrates the existence of a saddle point.

The intuition behind the theorem is in the name "mountain pass."

Then we know two low spots in the landscape: the origin because

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.

Note that this mountain pass is almost always a saddle point.

The assumptions of the theorem are: In this case there is a critical point

Moreover, if we define then For a proof, see section 5.5 of Aubin and Ekeland.