Qualitative analysis of positive radial singular solutions on hyperbolic space
Abstract
We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation ΔℍN u+λu+up=0 in ℍN∖{Q}, where N≥ 3, p>1, λ≤ ((N−1)2)/4, and Q∈ℍN is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near {Q} interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For N/(N−2)≤ p<(N+2)/(N−2), we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when p=N/(N−2), and the standard power-law profile when N/(N−2)<p<(N+2)/(N−2). At the conformally invariant pair (p,λ)=((N+2)/(N−2),(N(N−2))/4), we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime p>(N+2)/(N−2) with λ≤( N(N−2))/4, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.
Related
- Positive mass theorems for singular asymptotically hyperbolic manifolds
- Structure of radial solutions to a Hénon type equation with exponential nonlinearity on the hyperbolic space
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- Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
- Asymptotic Behavior of Radial Solutions to Singular p-Laplacian Equations as p→1
BibTeX
@misc{huang2026qualitative,
title = {{Qualitative analysis of positive radial singular solutions on hyperbolic space}},
author = {Xia Huang and Yahui Jiang and Chunyi Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06113},
eprint = {2609.06113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06113},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Qualitative analysis of positive radial singular solutions on hyperbolic space
AU - Xia Huang
AU - Yahui Jiang
AU - Chunyi Zhao
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06113
UR - https://arxiv.org/abs/2609.06113
AB - We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation Δ_ℍ^N u+λu+u^p=0 in ℍ^N∖{Q}, where N≥ 3, p>1, λ≤ ((N−1)^2)/4, and Q∈ℍ^N is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near {Q} interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For N/(N−2)≤ p<(N+2)/(N−2), we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when p=N/(N−2), and the standard power-law profile when N/(N−2)<p<(N+2)/(N−2). At the conformally invariant pair (p,λ)=((N+2)/(N−2),(N(N−2))/4), we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime p>(N+2)/(N−2) with λ≤( N(N−2))/4, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.
ER -