= Singular homogeneous solution of the Lane-Emden equation
{c}
Let $d\ge3$, $p>1$ and $a=2/(p-1)<d-2$. Then the <Generalized Lane-Emden equation> has the positive singular solution
$$
u_\infty(r)=Cr^{-a},\qquad C=[a(d-2-a)]^{1/(p-1)}.
$$
Indeed, 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 $v=C$.
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