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.

The assumptions of the theorem are:

If we define:

and:

then the conclusion of the theorem is that c is a critical value of I.

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 ${\displaystyle I[0]=0}$, and a far-off spot v where ${\displaystyle I[v]\leq 0}$. In between the two lies a range of mountains (at ${\displaystyle \Vert u\Vert =r}$) 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.

...
Wikipedia