Seek a <radial function> $u=u(r)$ on the unit ball. The equation and regularity at the origin give
$$
\left(r^{n-1}|u'|^{p-2}u'\right)'=r^{n-1},
\qquad
r^{n-1}|u'|^{p-2}u'=\frac{r^n}{n}.
$$
Hence $u'(r)=(r/n)^{1/(p-1)}$, and the zero <Dirichlet boundary condition> gives
$$
u(r)=\frac{p-1}{p}n^{-1/(p-1)}
\left(r^{p/(p-1)}-1\right).
$$
The power $|x|^q$ is twice differentiable at the origin exactly when $q\geq2$. Here $q=p/(p-1)$, so $u\in C^2$ at the origin exactly when $1<p\leq2$. Under the assumption $2<p<n$, this weak solution is never $C^2$ at the origin, exhibiting the regularity loss caused by <degenerate ellipticity>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 2 / d / Solution

{title2=$f(x)=g(|x|)$}
{wiki}

A radial function depends only on the distance from the origin: $f(x)=g(|x|)$. Differential operators then reduce to radial formulas such as $\Delta f=g''+(n-1)g'/r$.


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