Minimizing an action that leads to a non-linear second order differential equation of the Emden-Fowler type

Question

I'm trying to solve the following problem inspired by physics:

I have a functional $S$ defined by $$ S[f] = \frac{1}{2} \int\limits_0^\infty dr \left( \frac{f'(r)^2}{r^2} + \frac{f(r)^4}{r^6} \right) $$ and I am looking for real functions $f(r)$ over the interval $r \in (0, \infty)$ that minimize $S$, with the boundary conditions:

• $f(0) = 0$,
• $f(\infty) = c$ for some constant $c \geq 0$.

The solution in the case $c = 0$ is simply $f(r) = 0$. When $c \neq 0$ the problem must have a solution but I am not able to make any progress:

• Numerically this is a boundary value problem and I don't know how to solve it efficiently. My attempt using a basic relaxation method does not seem to work.
• Analytically, using the variational principle, $f$ must satisfy a second-order differential equation that is of the Emden-Fowler type: defining $t = r^3$ and $f(r) = g(t)$, the functional can be rewritten $$ S[g] = \frac{3}{2} \int\limits_0^\infty dt \left( g'(t)^2 + \frac{g(t)^4}{9 t^{8/3}} \right) $$ and the variational principle give $$ g''(t) = \frac{2}{9} \frac{g(t)^3}{t^{8/3}}, $$ but it does not seem to have closed-form solution (see Polyanin, A. D.; Zaitsev, Valentin F., Handbook of exact solutions for ordinary differential equations., Boca Raton, FL: CRC Press. xxvi, 787 p. (2003). ZBL1015.34001.).

Does anyone have a hint at how to tackle this problem?

Answer 1

I'm afraid that my question is not a very interesting one. It turns out that the problem does not admit non-trivial solutions.

To see this, split the integral into two pieces: $$ S[f] = S_1[f] + S_2[f] \quad \text{where} \quad S_1[f] = \frac{1}{2} \int\limits_0^\infty dr \, \frac{f'(r)^2}{r^2} \quad \text{and} \quad S_2[f] = \frac{1}{2} \int\limits_0^\infty dr \, \frac{f(r)^4}{r^6} $$ Each piece is separately non-negative, $S_1[f] \geq 0$ and $S_2[f] \geq 0$.

Now assume that we found a solution $f(r)$ minimizing $S[f]$, and define $f_\lambda(r) = f(\lambda r)$. Then we have $$ S[f_\lambda] = \lambda^3 S_1[f] + \lambda^5 S_2[f] $$ This implies $$ \frac{d}{d\lambda}S[f_\lambda] \Big|_{\lambda = 1} = 3 S_1[f] + 5 S_2[f] \geq 0 $$ By assumption that $f$ is a function minimizing $S[f]$, the derivative must be zero at $\lambda = 1$. This means that both $S_1[f] = 0$ and $S_2[f] = 0$, which is only satisfied by the trivial solution $f = 0$.

(Note that this proof is a variant of "Derrick's theorem")