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: If we define:
 * $$I$$ 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 $$I[v]\leq 0$$.
 * $$\Gamma=\{\mathbf{g}\in C([0,1];H)\,\vert\,\mathbf{g}(0)=0,\mathbf{g}(1)=v\}$$

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 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 $$\{x\in X\mid m(r)\le\Phi(x)\}$$.

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.