Mountain pass theorem

Remarks

Every path from $0$ to $e$ crosses the sphere $\norm{u} = r$, so its maximum is at least $a$, which gives $c \ge a$. Suppose $c$ were not a critical value. Then a deformation lemma, which uses the Palais–Smale condition, would push the sublevel set $\set{I \le c + \varepsilon}$ into $\set{I \le c - \varepsilon}$ while fixing $0$ and $e$. This would produce a path with maximum below $c$. The Palais–Smale condition cannot be dropped, even in $\R^2$. The function $f(x, y) = x^2 + (1 - x)^3 y^2$ has a strict local minimum at the origin with value $0$, so it is at least some $a > 0$ on a small circle around it. It also takes negative values, for example at $(2, 3)$, but its only critical point is the origin. Indeed, $\partial_y f = 2(1 - x)^3 y$ vanishes only for $x = 1$ or $y = 0$; at $x = 1$ one has $\partial_x f = 2$, and for $y = 0$ one needs $x = 0$. Antonio Ambrosetti and Paul Rabinowitz proved the theorem. A typical application gives a nonzero weak solution of $-\Delta u = \abs{u}^{p-1} u$ with zero boundary values on a bounded domain of $\R^n$, for $1 < p < (n+2)/(n-2)$ when $n \ge 3$.

Going deeper

For a student

The theorem is a basic tool of critical point theory and of variational methods for nonlinear elliptic equations. It finds solutions that are neither minima nor maxima of the energy, but saddle points. For semilinear problems, $0$ is often a trivial solution and a local minimum, and the theorem gives a second, nontrivial solution. It is the starting point of minimax methods, such as the linking theorems and Lusternik–Schnirelmann theory.