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  "body": "\n## Statement\n\nLet $X$ be a real [[Banach space|06GZ956657]] and let $I \\colon X \\to \\R$ be a continuously [[Fréchet differentiable|06EH97V2A7]] functional that satisfies the [[Palais–Smale condition|4724KRNHAT]].\nAssume that:\n\n1. $I(0) = 0$;\n2. there are $r \u003e 0$ and $a \u003e 0$ such that $I(u) \\ge a$ for every $u \\in X$ with $\\norm{u} = r$;\n3. there is $e \\in X$ with $\\norm{e} \u003e r$ and $I(e) \\le 0$.\n\nLet $\\Gamma$ be the set of continuous paths $\\gamma \\colon [0, 1] \\to X$ with $\\gamma(0) = 0$ and $\\gamma(1) = e$, and let\n$$c = \\inf_{\\gamma \\in \\Gamma} \\max_{t \\in [0, 1]} I(\\gamma(t)).$$\nThen $c \\ge a$, and $c$ is a critical value of $I$: there is $u \\in X$ with $I(u) = c$ and $I'(u) = 0$.\nIn particular $u$ is a [[critical point|0YTCMFB4HC]] of $I$ different from $0$ and from $e$.\n\n## Remarks\n\nEvery path from $0$ to $e$ crosses the sphere $\\norm{u} = r$, so its maximum is at least $a$, which gives $c \\ge a$.\nSuppose $c$ were not a critical value.\nThen 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 $e$.\nThis would produce a path with maximum below $c$.\nThe Palais–Smale condition cannot be dropped, even in $\\R^2$.\nThe 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 $a \u003e 0$ on a small circle around it.\nIt also takes negative values, for example at $(2, 3)$, but its only critical point is the origin.\nIndeed, $\\partial_y f = 2(1 - x)^3 y$ vanishes only for $x = 1$ or $y = 0$; at $x = 1$ one has $\\partial_x f = 2$, and for $y = 0$ one needs $x = 0$.\nAntonio Ambrosetti and Paul Rabinowitz proved the theorem.\nA 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 $1 \u003c p \u003c (n+2)/(n-2)$ when $n \\ge 3$.\n\n## Explanations\n\n### L1 human\n\nPicture a valley surrounded by a ring of mountains, with lower ground somewhere beyond the ring.\nAny route from the valley to that lower ground has to climb, and the best route goes through the lowest mountain pass.\nThe theorem says that this pass is a point where the ground is flat, that is, a critical point.\nFor functionals on infinite-dimensional spaces one needs a compactness condition, the Palais–Smale condition, to make the argument work.\n\n### L2 human\n\nThe theorem is a basic tool of critical point theory and of variational methods for nonlinear elliptic equations.\nIt finds solutions that are neither minima nor maxima of the energy, but saddle points.\nFor semilinear problems, $0$ is often a trivial solution and a local minimum, and the theorem gives a second, nontrivial solution.\nIt is the starting point of minimax methods, such as the linking theorems and Lusternik–Schnirelmann theory.\n",
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