Seek a homogeneous solution $u(r)=cr^{-\beta}$. Matching the powers in $\Delta u+u^p=0$ gives $\beta p=\beta+2$, hence
$$
\beta=\frac2{p-1}=a.
$$
Since
$$
\Delta r^{-\beta}=-\beta(d-2-\beta)r^{-\beta-2},
$$
matching coefficients gives
$$
c^{p-1}=\beta(d-2-\beta).
$$
Thus the positive <Singular homogeneous solution of the Lane-Emden equation> is
$$
u_\infty(r)=c_\infty r^{-2/(p-1)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 3 / Solution

{c}

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution
$$
u_\infty(r)=\bigl(a(d-2-a)\bigr)^{1/(p-1)}r^{-a}.
$$

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 Incoming links: Singular homogeneous solution of the Lane-Emden equation