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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.

PutThen $u=r^{-a}v$ and direct differentiation givesSince $ap=a+2$ and $d-2-2a=2(s_c-1)$, the Emden-Fowler transformation turns the equation intowhere

When $s_c=1$, one has $p=(d+2)/(d-2)$. The normalized Aubin-Talenti bubblesolves the radial equation, has $u(0)=1$, and tends to zero. It also follows from the stated one-dimensional soliton after the Emden-Fowler transformation, because the damping term vanishes.

As $r\to \infty$,so $\alpha =d-2$. Moreover $u^{\prime }(r)=O(r^{-(d-1)})$ at infinity and $u^{\prime }(r)=O(r)$ near zero. Thereforeand $\nabla u\in L^2({\mathbb{R}}^d)$.

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 $v=C$.