Classification of radially symmetric solutions to a Hardy-Hénon equation with sublinear absorption and applications
Abstract
Uniqueness of a non-negative radially symmetric solution with non-increasing profile to the Hardy-Hénon elliptic equation with sublinear absorption Δv(x) + |x|σv(x) - v1/m(x)=0, x∈ℝN, is established in the range N≥1, m>1 and σ∈(max{-2,-N},0). Consequences of this result are the uniqueness (up to scaling) of the non-negative extremal function to a Caffarelli-Kohn-Nirenberg inequality and the uniqueness of a non-negative radially symmetric self-similar solution with nonincreasing profile to a parabolic Hardy-Hénon equation with quasilinear diffusion. The proof relies on ordinary differential equations techniques, which can be used thanks to the radially symmetric setting, and requires in particular a delicate analysis of the well-posedness and continuous dependence in view of the non-Lipschitz nonlinearity and the singularity of |x|σ at the origin.
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BibTeX
@misc{iagar2026classification,
title = {{Classification of radially symmetric solutions to a Hardy-Hénon equation with sublinear absorption and applications}},
author = {Razvan Gabriel Iagar and Philippe Laurençot},
howpublished = {HAL},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06276},
eprint = {2611.06276},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05850287v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of radially symmetric solutions to a Hardy-Hénon equation with sublinear absorption and applications
AU - Razvan Gabriel Iagar
AU - Philippe Laurençot
PY - 2026
DA - 2026/11/03
PB - HAL
DO - 10.48550/arxiv.2611.06276
UR - https://hal.science/hal-05850287v1
AB - Uniqueness of a non-negative radially symmetric solution with non-increasing profile to the Hardy-Hénon elliptic equation with sublinear absorption Δv(x) + |x|σv(x) - v1/m(x)=0, x∈ℝN, is established in the range N≥1, m>1 and σ∈(max{-2,-N},0). Consequences of this result are the uniqueness (up to scaling) of the non-negative extremal function to a Caffarelli-Kohn-Nirenberg inequality and the uniqueness of a non-negative radially symmetric self-similar solution with nonincreasing profile to a parabolic Hardy-Hénon equation with quasilinear diffusion. The proof relies on ordinary differential equations techniques, which can be used thanks to the radially symmetric setting, and requires in particular a delicate analysis of the well-posedness and continuous dependence in view of the non-Lipschitz nonlinearity and the singularity of |x|σ at the origin.
ER -