mountain pass theorem

Let $X$ a real Banach space and $F\in C^1(X,ℝ)$. Consider $K$ a compact metric space, and $K^{*}\subset K$ a closed nonempty subset of $K$. If $p^{*}:K^{*}\to X$ is a continuous mapping, set

$$𝒫=\{p\in C(K,X);p=p^{*}\text{on }{K}^{*}\}.$$

Define

$$c=\underset{p\in 𝒫}{inf}\underset{t\in K}{\max }F(p(t)).$$

Assume that

$$c>\underset{t\in K^{*}}{\max }F(p^{*}(t)).$$(1)

Then there exists a sequence $(x_n)$ in $X$ such that

1. $\underset{n\to \mathrm{\infty }}{lim}F(x_n)=c$;

2. $\underset{n\to \mathrm{\infty }}{lim}\parallel F^{\prime }(x_n)\parallel =0$.

The name of this theorem is a consequence of a simplified visualization for the objects from theorem. If we consider the set $K^{*}=\{A,B\}$, where $A$ and $B$ are two villages, $𝒫$ is the set of all the routes from $A$ to $B$, and $F(x)$ represents the altitude of point $x$; then the assumption (1) is equivalent to say that the villages $A$ and $B$ are separated with a mountains chain. So, the conclusion of the theorem tell us that exists a route between the villages with a minimal altitude. With other words exists a “mountain pass” .