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:

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

If we define:

$\mathrm{\Gamma }=\{\mathbf{𝐠}\in C([0,1];H)|\mathbf{𝐠}(0)=0,\mathbf{𝐠}(1)=v\}$

and:

$c=\underset{\mathbf{𝐠}\in \mathrm{\Gamma }}{\inf }\underset{0\le t\le 1}{\max }I[\mathbf{𝐠}(t)],$

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

• $\mathrm{\Phi }\in C(X,\mathbf{𝐑})$ and have a Gateaux derivative ${\mathrm{\Phi }}^{\prime }:X\to X^{*}$ which is continuous when $X$ and $X^{*}$ are endowed with strong topology and weak* topology respectively.
• There exists $r>0$ such that one can find certain $\Vert x^{\prime }\Vert >r$ with

$\max (\mathrm{\Phi }(0),\mathrm{\Phi }(x^{\prime }))<\inf {}_{\Vert x\Vert =r}\mathrm{\Phi }(x)=:m(r)$.

• $\mathrm{\Phi }$ satisfies weak Palais–Smale condition on $\{x\in X\mid m(r)\le \mathrm{\Phi }(x)\}$.

In this case there is a critical point $\bar{x}\in X$ of $\mathrm{\Phi }$ satisfying $m(r)\le \mathrm{\Phi }(\bar{x})$. Moreover, if we define

$\mathrm{\Gamma }=\{c\in C([0,1],X)\mid c(0)=0,c(1)=x^{\prime }\}$

then

$\mathrm{\Phi }(\bar{x})=\underset{c\in \mathrm{\Gamma }}{\inf }\underset{0\le t\le 1}{\max }\mathrm{\Phi }(c(t)).$

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.