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The **mountain pass theorem** is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.^{[1]} 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.

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*.

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.

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.

- Aubin, Jean-Pierre; Ekeland, Ivar (2006).
*Applied Nonlinear Analysis*. Dover Books. ISBN 0-486-45324-3. - Bisgard, James (2015). "Mountain Passes and Saddle Points".
*SIAM Review*.**57**(2): 275–292. doi:10.1137/140963510. - Evans, Lawrence C. (1998).
*Partial Differential Equations*. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2. - Jabri, Youssef (2003).
*The Mountain Pass Theorem, Variants, Generalizations and Some Applications*. Encyclopedia of Mathematics and its Applications. Cambridge University Press. ISBN 0-521-82721-3. - Mawhin, Jean; Willem, Michel (1989). "The Mountain Pass Theorem and Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems".
*Critical Point Theory and Hamiltonian Systems*. New York: Springer-Verlag. pp. 92–97. ISBN 0-387-96908-X. - McOwen, Robert C. (1996). "Mountain Passes and Saddle Points".
*Partial Differential Equations: Methods and Applications*. Upper Saddle River, NJ: Prentice Hall. pp. 206–208. ISBN 0-13-121880-8.