 Singular homogeneous solution of the 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 $v=C$.