Critical Points and the Mountain Pass

Minimisation finds the lowest point of an energy. But many solutions of physical and geometric problems are not minima: they are saddle points, critical points that are minima in some directions and maxima in others. The narrow, unstable catenoid of 6A.11 Calculus of Variations and Gradient Flows is one; a closed geodesic on a sphere is another (every closed curve can be shrunk to a point, so the shortest closed curve has length zero, yet great circles exist); the unstable minimal surfaces that Colding and Minicozzi use to prove that the Ricci flow becomes extinct are a third (12C.3 Finite Extinction). To find saddle points one needs min–max methods: take a family of paths (or surfaces, or sweepouts) joining two low regions, maximise the energy along each, and minimise the maximum over the family. If the energy landscape has a "mountain range" between the two regions, the min–max value is the height of the lowest pass, and the pass is a critical point.

Making this rigorous in infinite dimensions needs a replacement for compactness, the Palais–Smale condition, and a way to push a path downhill, the deformation lemma. With these, the mountain pass theorem of Ambrosetti and Rabinowitz (1973) produces critical points, and its standard application gives non-trivial solutions of semilinear elliptic equations such as $-\Delta u = |u|^{p-1}u$. The method fails exactly at the critical Sobolev exponent $p = \frac{n+2}{n-2}$, where compactness is lost to concentrating bubbles (4A.11 Sobolev Embeddings and Critical Exponents); Pohozaev's identity shows that on star-shaped domains there are then no solutions at all. That borderline is the analytic heart of the Yamabe problem.

By the end of this chapter you will be able to:

• define critical points and critical values of a $C^1$ functional on a Hilbert space, and the Palais–Smale condition;
• prove the deformation lemma by flowing down a cut-off gradient field;
• prove the mountain pass theorem, and explain with an example why Palais–Smale cannot be dropped;
• prove that $-\Delta u = |u|^{p-1}u$ has a non-trivial solution on any bounded domain for $1 < p < \frac{n+2}{n-2}$;
• prove Pohozaev's identity and deduce non-existence on star-shaped domains for $p \geq \frac{n+2}{n-2}$.

2.1Mountain passes in chemistry

2.2Critical points and the Palais–Smale condition

Let $H$ be a real Hilbert space and $I : H \to \mathbb{R}$ a functional of class $C^1$: at each $u$ there is a bounded linear map $I'(u)$ with $I(u + v) = I(u) + I'(u)v + o(\|v\|)$, depending continuously on $u$. By Riesz (4A.5 Hilbert Spaces and Lax–Milgram) we identify $I'(u)$ with a vector, the gradient. A point $u$ is a critical point if $I'(u) = 0$, and $c$ is a critical value if $c = I(u)$ for some critical point $u$. We assume throughout, as Evans does, that $I'$ is Lipschitz on bounded sets; this makes the ODEs below well posed.

In finite dimensions, a continuous function on a compact set attains its extrema, and that is what makes min–max values critical. In infinite dimensions, bounded sets are not compact (4A.1 Banach Spaces and Bounded Operators), and a "mountain pass" can fail to have a critical point because the near-critical points run off to infinity. The condition that excludes this is:

A sequence with $I(u_k)$ bounded and $I'(u_k) \to 0$ is a Palais–Smale sequence. If (PS) holds, the limit of a convergent subsequence is a critical point (by continuity of $I'$). Verifying (PS) for a given functional is usually the main analytic work, and it typically uses a compact Sobolev embedding.

2.3The deformation lemma

The idea is to flow down the gradient. Where the gradient is not small, the flow lowers the energy at a definite rate. The deformation lemma says: if $c$ is not a critical value, then the sublevel set $\{I \leq c + \varepsilon\}$ can be continuously pushed into $\{I \leq c - \varepsilon\}$.

The cut-off $g$ freezes points far from the level $c$, which is what keeps the endpoints of a path in place below; the normalisation $h$ keeps the flow from moving infinitely far in finite time. Everything else is "flow downhill".

2.4The mountain pass theorem

In words: if no critical point were at the level $c$, a nearly optimal path could be pushed down below $c$, which is impossible. The critical point found is typically a saddle: it is at least as high as the mountain range, and every path to $v$ must go at least as high as it.

Palais–Smale cannot be dropped. Without it, the min–max value need not be critical, even in two dimensions. The function $f(x, y) = x^2 + (1 - x)^3y^2$ on $\mathbb{R}^2$ has a strict local minimum at the origin, takes negative values (for example at $(2, 3)$), and so has mountain-pass geometry, but its only critical point is the origin (Exercise 2.7). The almost-critical points escape to infinity along the valley; (PS) fails.

2.5A semilinear equation

Let $U \subset \mathbb{R}^n$ be bounded, $n \geq 3$, and $1 < p < \frac{n+2}{n-2}$. We look for non-trivial solutions of

The zero function is a solution, and the equation is the Euler–Lagrange equation of

with the inner product $\langle u, v\rangle = \int\nabla u\cdot\nabla v$. The exponent condition says $p + 1 < \frac{2n}{n-2} = 2^*$, so $H^1_0 \hookrightarrow L^{p+1}$ is continuous (Sobolev) and compact (Rellich–Kondrachov, 4A.11 Sobolev Embeddings and Critical Exponents).

Applying the theorem to $I_+[u] = \int\frac12|\nabla u|^2 - \frac{1}{p+1}(u^+)^{p+1}$ instead produces a non-negative solution, which is positive by the strong maximum principle. The same solutions were found as constrained minimisers in 6C.1 Regularity of Minimisers and Constraints; the mountain pass gives more, in that the method extends to non-homogeneous nonlinearities $f(u)$ that are superlinear at $0$ and at infinity with the Ambrosetti–Rabinowitz condition $0 < \theta F(u) \leq uf(u)$ for some $\theta > 2$, which is what made the coefficient positive in the boundedness step.

2.6The critical exponent and Pohozaev's identity

Where did $p < \frac{n+2}{n-2}$ enter? Only in the compactness of $H^1_0 \hookrightarrow L^{p+1}$. At $p = \frac{n+2}{n-2}$ the embedding into $L^{2^*}$ is not compact, because of concentrating bubbles (4A.11 Sobolev Embeddings and Critical Exponents), and (PS) fails: rescaled bubbles $\varepsilon^{-(n-2)/2}w(x/\varepsilon)$ form Palais–Smale sequences with no convergent subsequence. Is this only a failure of the method? On star-shaped domains it is a genuine non-existence, by an identity that comes from the scaling symmetry of the equation.

The critical exponent is where the Pohozaev identity's two terms balance, and it is the exponent at which the equation is invariant under the scaling $u(x) \mapsto \lambda^{(n-2)/2}u(\lambda x)$, which preserves $\int|\nabla u|^2$. Scaling invariance is what destroys compactness (a bubble can shrink without changing its energy), and, through Pohozaev, what forbids solutions on star-shaped domains. On domains with holes (annuli), solutions do exist at the critical exponent (Kazdan and Warner 1975 for annuli by symmetry; Bahri and Coron 1988 in general for domains with non-trivial topology); and Haim Brezis and Louis Nirenberg showed in 1983 that adding a lower-order term $\lambda u$ restores existence for suitable $\lambda$ in dimensions $n \geq 4$.

2.7History

Marston Morse developed critical point theory in finite dimensions and on spaces of curves in the 1920s and 30s, and George Birkhoff's min–max argument for closed geodesics on the sphere dates from 1917. Richard Palais and Stephen Smale introduced their condition in 1963–64, to extend Morse theory to infinite-dimensional manifolds. Antonio Ambrosetti and Paul Rabinowitz proved the mountain pass theorem in 1973. Stanislav Pohozaev's identity appeared in 1965; physicists know its relative as Derrick's theorem (1964) on the non-existence of stable static solitons in field theories. Haim Brezis and Louis Nirenberg's critical-exponent paper appeared in 1983. Eyring, Evans and Polanyi's transition state theory dates from 1935, and the nudged elastic band method from 1998.

2.8Exercises

Solution

(a) Near the origin $(1 - x)^3 \approx 1$, so $f \approx x^2 + y^2$, positive except at $0$: a strict local minimum, and on a small circle $f \geq a > 0$. And $(1 - 2)^3 = -1$, so $f(2, 3) = 4 - 9 = -5$. (b) $\partial_yf = 2(1 - x)^3y = 0$ forces $y = 0$ or $x = 1$. If $y = 0$, $\partial_xf = 2x = 0$ gives the origin. If $x = 1$, $\partial_xf = 2x - 3(1 - x)^2y^2 = 2 \neq 0$. (c) If (PS) held, the theorem would give a critical point with value $c \geq a > 0$; the only critical point has value $0$. So (PS) fails: there are Palais–Smale sequences at the level $c$ with no convergent subsequence, which must escape to infinity (a bounded sequence in $\mathbb{R}^2$ has a convergent subsequence, whose limit would be critical). The near-optimal paths go out to infinity along the valley where $(1 - x)^3y^2$ is very negative, and the "pass" is never reached.

Solution

$e^{-u}$: $I'(u) = -e^{-u} \neq 0$; $u_k = k$ gives $I \to 0$, $I' \to 0$, no convergent subsequence. $u^2(1 - u)^2$: (PS) holds because $I(u_k)$ bounded forces $u_k$ bounded ($I \to \infty$ as $|u| \to \infty$), and bounded sequences in $\mathbb{R}$ have convergent subsequences. $I(0) = I(1) = 0$ and $I > 0$ in between; the mountain pass level is $\max_{[0,1]}I = I(\frac12) = \frac{1}{16}$, at the critical point $u = \frac12$.

Solution

(a) $5$, $3$, $\frac73$. (b) $\nabla u_\lambda = \lambda^{n/2}(\nabla u)(\lambda x)$, so $\int|\nabla u_\lambda|^2 = \lambda^n\lambda^{-n}\int|\nabla u|^2$. $-\Delta u_\lambda = \lambda^{(n+2)/2}(-\Delta u)(\lambda x)$ and $|u_\lambda|^{p-1}u_\lambda = \lambda^{p(n-2)/2}(|u|^{p-1}u)(\lambda x)$; equal for all $\lambda$ iff $\frac{n+2}{2} = \frac{p(n-2)}{2}$. (c) Negative for $p < \frac{n+2}{n-2}$, zero at equality, positive above.

Solution

With $\int|\nabla u|^2 = \lambda\int u^2$ and $\int F = \frac\lambda2\int u^2$: $\frac{n-2}{2}\lambda\int u^2 - \frac{n\lambda}{2}\int u^2 = -\lambda\int u^2 = -\frac12\int_{\partial B}(\partial_\nu u)^2$. For $n = 1$: $u' = -\frac\pi2\sin\frac{\pi x}{2}$, so $(u')^2 = \frac{\pi^2}{4}$ at $x = \pm1$, and the boundary "integral" is the sum over the two points, $\frac{\pi^2}{2}$; the right side is $2\cdot\frac{\pi^2}{4}\int_{-1}^1\cos^2\frac{\pi x}{2}dx = \frac{\pi^2}{2}\cdot1$. They agree.

Solution

$I(u_k) - \frac1\theta I'(u_k)u_k = \big(\frac12 - \frac1\theta\big)\|u_k\|^2 + \int\big(\frac1\theta u_kf(u_k) - F(u_k)\big)$, and the integral is bounded below (the integrand is $\geq 0$ for large $|u_k|$ and bounded for small). The left side is at most $C + o(1)\|u_k\|$. So $\|u_k\|$ is bounded. For $F = \frac12s^2$, $\theta = 2$ and the coefficient $\frac12 - \frac1\theta$ is $0$: the method gives nothing, consistent with the fact that $-\Delta u = u$ has non-trivial solutions only when $1$ is an eigenvalue, and then a whole line of them.

Solution

(a) $\frac{d}{dt}\big((t + C)^{-3/4}W\big) = (t + C)^{-3/4}\big(W' - \frac{3}{4(t+C)}W\big) \leq -4\pi(t + C)^{-3/4}$. (b) Integrate from $0$ to $t$: $\int_0^t(s + C)^{-3/4}ds = 4\big((t + C)^{1/4} - C^{1/4}\big)$. (c) The right side tends to $-\infty$, so $W(t) < 0$ for large $t$ if the flow existed that long; but $W \geq 0$ (indeed $> 0$) for every smooth metric, so the flow must become singular, and on these manifolds the singularity is extinction.