Fundamental solutions of the fractional p-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity
Abstract
Let $ C_ω$ be the cone generated by a proper C2 domain ω⊂𝕊N−1. We study positive homogeneous solutions of the fractional p-Laplace equation in Cω with zero exterior values. For p≠2, the operator is nonlinear and not uniformly elliptic. For every p>1, we identify two extremal homogeneities β+>0>β-, prove that every fundamental solution, if one exists, has one of these exponents, and construct homogeneous eigenfunctions whose eigenvalues have signs determined by the position of the homogeneity relative to β±. As an application, we obtain the Lane-Emden threshold q*=p−1+sp/β+ for every p>1: nontrivial nonnegative supersolutions of (−Δp)s u≥ uq in Cω do not exist for 0<q<q*, whereas homogeneous supersolutions exist for q>q*. We also establish Hopf-type lower bounds at conical boundary points with the optimal exponent. When p−1>sp , we construct both fundamental solutions, one singular at the vertex and the other vanishing there. For p=2, these solutions are known; our method recovers them for every s∈(0,1). The homogeneous eigenfunctions also serve as barriers, yielding boundary decay and global Hölder regularity for (−Δp)s u=f in bounded Lipschitz domains, with zero exterior values and f continuous up to the boundary. The admissible Hölder exponents are determined by the exterior cones. All results hold for every N≥2; in particular, N>sp is not needed.
Related
- Hölder regularity of solutions to parabolic equations associated with fractional p-Laplacian type operators
- Sharp regularity for the anisotropic fractional Laplacian with Morrey data
- Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
- Tempered fractional Laplacian: Boundary regularity
- Large Solutions for Fractional Laplacian on Infinite Cylindrical Domains
BibTeX
@misc{biswas2026fundamental,
title = {{Fundamental solutions of the fractional $p$-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity}},
author = {Reshmi Biswas and Alexander Quaas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07157},
eprint = {2611.07157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07157},
}
RIS
TY - UNPB
TI - Fundamental solutions of the fractional p-Laplacian in cone-like domains, with applications to Liouville-type theorems and global Hölder regularity
AU - Reshmi Biswas
AU - Alexander Quaas
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07157
UR - https://arxiv.org/abs/2611.07157
AB - Let $ C_ω$ be the cone generated by a proper C^2 domain ω⊂𝕊^N−1. We study positive homogeneous solutions of the fractional p-Laplace equation in C_ω with zero exterior values. For p≠2, the operator is nonlinear and not uniformly elliptic. For every p>1, we identify two extremal homogeneities β^+>0>β^-, prove that every fundamental solution, if one exists, has one of these exponents, and construct homogeneous eigenfunctions whose eigenvalues have signs determined by the position of the homogeneity relative to β^±. As an application, we obtain the Lane-Emden threshold q^*=p−1+sp/β^+ for every p>1: nontrivial nonnegative supersolutions of (−Δ_p)^s u≥ u^q in C_ω do not exist for 0<q<q^*, whereas homogeneous supersolutions exist for q>q^*. We also establish Hopf-type lower bounds at conical boundary points with the optimal exponent. When p−1>sp , we construct both fundamental solutions, one singular at the vertex and the other vanishing there. For p=2, these solutions are known; our method recovers them for every s∈(0,1). The homogeneous eigenfunctions also serve as barriers, yielding boundary decay and global Hölder regularity for (−Δ_p)^s u=f in bounded Lipschitz domains, with zero exterior values and f continuous up to the boundary. The admissible Hölder exponents are determined by the exterior cones. All results hold for every N≥2; in particular, N>sp is not needed.
ER -