When $s_c=1$, one has $p=(d+2)/(d-2)$. The normalized <Aubin-Talenti bubble>
$$
u(r)=\left(1+\frac{r^2}{d(d-2)}\right)^{-(d-2)/2}
$$
solves 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 $r\to\infty$,
$$
r^{d-2}u(r)\longrightarrow[d(d-2)]^{(d-2)/2}>0,
$$
so $\alpha=d-2$. Moreover $u'(r)=O(r^{-(d-1)})$ at infinity and $u'(r)=O(r)$ near zero. Therefore
$$
\int_{\mathbb R^d}|\nabla u|^2,dx
\asymp\int_0^\infty|u'(r)|^2r^{d-1},dr<\infty,
$$
and $\nabla u\in L^2(\mathbb R^d)$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 1 / 1 / 4 / Solution

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{wiki=Sobolev_inequality#Extremal_functions}

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubble
$$
W(x)=\left(1+\frac{|x|^2}{d(d-2)}\right)^{-(d-2)/2}
$$
solves $\Delta W+W^p=0$ and optimizes the critical <Sobolev embedding theorem>.

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 Incoming links: Aubin-Talenti bubble