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## Summary

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 $I'$  is Lipschitz continuous on bounded subsets of H,
• $I$  satisfies the Palais–Smale compactness condition,
• $I=0$ ,
• 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 $I=0$ , 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 Gateaux 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 $\{x\in X\mid m(r)\leq \Phi (x)\}$ .

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

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

then

$\Phi ({\overline {x}})=\inf _{c\,\in \,\Gamma }\max _{0\leq t\leq 1}\Phi (c\,(t)).$

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