Isolated singularities of weak solutions to the Serrin-critical $p$-Laplace equation

We classify isolated singularities of nonnegative weak solutions to the Serrin-critical quasilinear Lane-Emden equation $-{\mathrm{\Delta }}_{p}u=u^{\frac{n(p-1)}{n-p}}$ in $B_1\setminus \{0\}$, where $1<p<n$, ${\mathrm{\Delta }}_p$ is the classical $p$-Laplacian operator and $B_1\setminus \{0\}\subset {\mathbb{R}}^n$ is the punctured unit ball. We prove that either the origin is a removable singularity or $u$ satisfies ${\lim }_{x\to 0}\mathrm{\mid }x{\mathrm{\mid }}^{\frac{n-p}{p-1}}(\log \frac{1}{\mathrm{\mid }x\mathrm{\mid }}{)}^{\frac{n-p}{p(p-1)}}u(x)=[\frac{n-p}{p}(\frac{n-p}{p-1}{)}^{p-1}{]}^{\frac{n-p}{p(p-1)}}$. This extends the result of Aviles (Comm. Math. Phys. 1987: 177–192) from the semilinear case $p=2$ to the full range $1<p<n$, without assuming radial symmetry. We develop a new BV-based argument for weak solutions. We also give a direct construction of positive radial singular solutions with the same asymptotic behavior.