Emden-Fowler transformation

The Emden-Fowler transformation turns radial scaling into translation in a logarithmic variable. For a radial Generalized Lane-Emden equationput $s=\log r$, $a=2/(p-1)$ and $u(r)=r^{-a}v(s)$. Direct differentiation givesThe primes in the second equation are derivatives with respect to $s$. The choice $ap=a+2$ makes every term carry the same power of $r$, leaving an autonomous ordinary differential equation.

A constant nonzero equilibrium $v$ becomes a singular power law $u(r)=C{r}^{-a}$, explaining the Singular homogeneous solution of the Lane-Emden equation. Phase-plane trajectories instead describe profiles whose behavior changes with radius. The transformation is useful because dilation of $r$ becomes a shift of $s$, which an autonomous equation handles naturally.

The classical Lane-Emden equation is the dimensionless equationfor a spherically symmetric polytrope in three dimensions. The Generalized Lane-Emden equation extends it to arbitrary dimensions and to functions without radial symmetry.

The regular dimensionless solution of the Lane-Emden equation with ${\theta }_n(0)=1$ and ${\theta }_n^{\prime }(0)=0$. Its local expansion is ${\theta }_n(\xi )=1-{\xi }^2/6+n{\xi }^4/120+O({\xi }^6)$. If it has a first zero ${\xi }_1$, that zero and $-{\xi }_1^2{\theta }_n^{\prime }({\xi }_1)$ specify radius and mass factors for the corresponding finite-radius stellar polytrope.

A Generalized Lane-Emden equation is a semilinear partial differential equation whose highest-order term is the Laplacian:For a radial function in $d$ dimensions it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$. The case $d=3$ gives the classical Lane-Emden equation.

Let $d\ge 3$, $p>1$ and $a=2/(p-1)<d-2$. Then the Generalized Lane-Emden equation has the positive singular solutionIndeed, the radial Laplacian satisfies $\Delta r^{-a}=a(a+2-d)r^{-a-2}$. Matching its exponent with $u^p$ gives $ap=a+2$; matching coefficients gives $C^{p-1}=a(d-2-a)>0$. The strict inequality on $a$ is exactly what makes that positive coefficient possible.

The solution is singular at the origin, so it is distinct from the regular Lane-Emden function normalized there. Under the Emden-Fowler transformation, it is the constant equilibrium $v=C$.

For $d\ge 3$, at the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.