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:

• [math]\displaystyle{ I }[/math] is a functional from a Hilbert space H to the reals,
• [math]\displaystyle{ I\in C^1(H,\mathbb{R}) }[/math] and [math]\displaystyle{ I' }[/math] is Lipschitz continuous on bounded subsets of H,
• [math]\displaystyle{ I }[/math] satisfies the Palais–Smale compactness condition,
• [math]\displaystyle{ I[0]=0 }[/math],
• there exist positive constants r and a such that [math]\displaystyle{ I[u]\geq a }[/math] if [math]\displaystyle{ \Vert u\Vert =r }[/math], and
• there exists [math]\displaystyle{ v\in H }[/math] with [math]\displaystyle{ \Vert v\Vert \gt r }[/math] such that [math]\displaystyle{ I[v]\leq 0 }[/math].

If we define:

[math]\displaystyle{ \Gamma=\{\mathbf{g}\in C([0,1];H)\,\vert\,\mathbf{g}(0)=0,\mathbf{g}(1)=v\} }[/math]

and:

[math]\displaystyle{ c=\inf_{\mathbf{g}\in\Gamma}\max_{0\leq t\leq 1} I[\mathbf{g}(t)], }[/math]

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 [math]\displaystyle{ I[0]=0 }[/math], and a far-off spot v where [math]\displaystyle{ I[v]\leq 0 }[/math]. In between the two lies a range of mountains (at [math]\displaystyle{ \Vert u\Vert =r }[/math]) 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 [math]\displaystyle{ X }[/math] be Banach space. The assumptions of the theorem are:

• [math]\displaystyle{ \Phi\in C(X,\mathbf R) }[/math] and have a Gateaux derivative [math]\displaystyle{ \Phi'\colon X\to X^* }[/math] which is continuous when [math]\displaystyle{ X }[/math] and [math]\displaystyle{ X^* }[/math] are endowed with strong topology and weak* topology respectively.
• There exists [math]\displaystyle{ r\gt 0 }[/math] such that one can find certain [math]\displaystyle{ \|x'\|\gt r }[/math] with

[math]\displaystyle{ \max\,(\Phi(0),\Phi(x'))\lt \inf\limits_{\|x\|=r}\Phi(x)=:m(r) }[/math].

• [math]\displaystyle{ \Phi }[/math] satisfies weak Palais–Smale condition on [math]\displaystyle{ \{x\in X\mid m(r)\le\Phi(x)\} }[/math].

In this case there is a critical point [math]\displaystyle{ \overline x\in X }[/math] of [math]\displaystyle{ \Phi }[/math] satisfying [math]\displaystyle{ m(r)\le\Phi(\overline x) }[/math]. Moreover, if we define

[math]\displaystyle{ \Gamma=\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x'\} }[/math]

then

[math]\displaystyle{ \Phi(\overline x)=\inf_{c\,\in\,\Gamma}\max_{0\le t\le 1}\Phi(c\,(t)). }[/math]

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

Further reading

• 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. http://epubs.siam.org/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. https://archive.org/details/mountainpasstheo0000jabr. 
• 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. https://books.google.com/books?id=w6bTBwAAQBAJ&pg=PA92. 
• 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. https://books.google.com/books?id=TuNHsNC1Yf0C&pg=PA206.