# Mountain pass theorem

The mountain pass theorem is an existence theorem from the calculus of variations. 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.

## Theorem 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 $I'$ 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 $I[u]\geq a$ if $\Vert u\Vert =r$, and
• there exists $v\in H$ with $\Vert v\Vert >r$ such that $I[v]\leq 0$.

If we define:

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

and:

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

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 $I[v]\leq 0$. 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:

• $\Phi\in C(X,\mathbf R)$ and have a Gâteaux derivative $\Phi'\colon X\to X^*$ 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 $\|x'\|>r$ with
$\max\,(\Phi(0),\Phi(x'))<\inf\limits_{\|x\|=r}\Phi(x)=:m(r)$.
• $\Phi$ satisfies weak Palais-Smale condition on $\{x\in X\mid m(r)\le\Phi(x)\}$.

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

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

then

$\Phi(\overline x)=\inf_{c\,\in\,\Gamma}\max_{0\le t\le 1}\Phi(c\,(t)).$

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

## References

• 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.
• Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2.
• Aubin, Jean-Pierre; Ivar Ekeland (2006). Applied Nonlinear Analysis. Dover Books. ISBN 0-486-45324-3.