PreprintarXiv licencearXiv
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 −Δp u=u(n(p−1))/(n−p) in B1∖{0}, where 1<p<n, Δp is the classical p-Laplacian operator and B1∖{0}⊂ℝn is the punctured unit ball. We prove that either the origin is a removable singularity or u satisfies limx→0|x|(n−p)/(p−1)(log1/|x|)(n−p)/(p(p−1))u(x)=[(n−p)/p((n−p)/(p−1))p−1](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.
Related
- Isolated singularities of solutions to the Yamabe equation in higher dimensions I
- Isolated singularities of solutions to the Yamabe equation in higher dimensions II
- The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II
- Ground state solutions for fractional p-Laplace equations with mixed nonlinearities in the critical and subcritical cases
- Normalized solutions to 1-Laplace equations with critical growth term
BibTeX
@misc{xu2026isolated,
title = {{Isolated singularities of weak solutions to the Serrin-critical $p$-Laplace equation}},
author = {Meiqing Xu and Hui Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23045},
eprint = {2610.23045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23045},
}
RIS
TY - UNPB
TI - Isolated singularities of weak solutions to the Serrin-critical p-Laplace equation
AU - Meiqing Xu
AU - Hui Yang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23045
UR - https://arxiv.org/abs/2610.23045
AB - We classify isolated singularities of nonnegative weak solutions to the Serrin-critical quasilinear Lane-Emden equation −Δ_p u=u^(n(p−1))/(n−p) in B_1∖{0}, where 1<p<n, Δ_p is the classical p-Laplacian operator and B_1∖{0}⊂ℝ^n is the punctured unit ball. We prove that either the origin is a removable singularity or u satisfies lim_x→0|x|^(n−p)/(p−1)(log1/|x|)^(n−p)/(p(p−1))u(x)=[(n−p)/p((n−p)/(p−1))^p−1]^(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.
ER -