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

Abstract

We classify isolated singularities of nonnegative weak solutions to the Serrin-critical quasilinear Lane-Emden equation $-Δ_p u=u^{\frac{n(p-1)}{n-p}}$ in $B_1\setminus\{0\}$, where $1<p<n$, $Δ_p$ 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 $\lim_{x\to0}|x|^{\frac{n-p}{p-1}}(\log\frac1{|x|})^{\frac{n-p}{p(p-1)}}u(x)=[\frac{n-p}{p}(\frac{n-p}{p-1})^{p-1}]^{\frac{n-p}{p(p-1)}}$. 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.