Emden-Fowler transformation

The Emden-Fowler transformation is a change of variables for scaling-invariant radial equations, including the Lane-Emden equation. The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

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.

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.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

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.