Revisiting Guo-Huang-Wang-Wei pointwise estimates for radial solutions to the fourth-order Hénon equation in the punctured space
Abstract
We study pointwise estimates for positive, C4-solution u to the fourth-order Hénon equation Δ2 u = |x|σup in Rn ∖ {0} with n ≥ 5, p>1, and $σ>-4 $. In an earlier work published in J. Anal. Math. 140 (2020) 371-394, Z. Guo, X. Huang, L. Wang, and J. Wei established the following pointwise estimate −Δu ≥ 2/(n−4) (|∇ u|2)/u + √((n(n−4))/(n2−4)) u/nn−4 in Rn ∖ {0} for radial solutions to the critical, fourth-order Lane-Emden equation, namely σ= 0 and p=(n+4)/(n−4). Inspired by this interesting result, we establish a similar pointwise inequality −Δu ≥ 2/(n−4) (|∇ u|2)/u + √(2/(p+1− 8/(n(n−4)))) |x|σ/2 u/p+12 in Rn ∖ {0} for radial solutions to the fourth-order Hénon equation in Rn ∖ {0} for any σ≥ 0 and p ≥ (n+4 + σ)/(n−4). We also show that the constant 2/(n−4) is sharp and that the thresholds σ≥ 0 and p ≥ (n+4 + σ)/(n−4) are necessary. As in our previous study on the case of Rn, our proof is also based on the Bernstein technique via the maximum principle, hence appears to be simpler and more constructive. However, to avoid possible singularity at the origin, we develop a Bernstein technique in one-dimensional space, which is capable of handling equations in punctured spaces.
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BibTeX
@misc{ngo2026revisiting,
title = {{Revisiting Guo-Huang-Wang-Wei pointwise estimates for radial solutions to the fourth-order Hénon equation in the punctured space}},
author = {Quoôc Anh Ngô and Trung Nguyên},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39134},
eprint = {2610.39134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39134},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Revisiting Guo-Huang-Wang-Wei pointwise estimates for radial solutions to the fourth-order Hénon equation in the punctured space
AU - Quoôc Anh Ngô
AU - Trung Nguyên
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39134
UR - https://arxiv.org/abs/2610.39134
AB - We study pointwise estimates for positive, C^4-solution u to the fourth-order Hénon equation Δ^2 u = |x|^σu^p in R^n ∖ {0} with n ≥ 5, p>1, and $σ>-4 $. In an earlier work published in J. Anal. Math. 140 (2020) 371-394, Z. Guo, X. Huang, L. Wang, and J. Wei established the following pointwise estimate −Δu ≥ 2/(n−4) (|∇ u|^2)/u + √((n(n−4))/(n^2−4)) u^/nn−4 in R^n ∖ {0} for radial solutions to the critical, fourth-order Lane-Emden equation, namely σ= 0 and p=(n+4)/(n−4). Inspired by this interesting result, we establish a similar pointwise inequality −Δu ≥ 2/(n−4) (|∇ u|^2)/u + √(2/(p+1− 8/(n(n−4)))) |x|^σ/2 u^/p+12 in R^n ∖ {0} for radial solutions to the fourth-order Hénon equation in R^n ∖ {0} for any σ≥ 0 and p ≥ (n+4 + σ)/(n−4). We also show that the constant 2/(n−4) is sharp and that the thresholds σ≥ 0 and p ≥ (n+4 + σ)/(n−4) are necessary. As in our previous study on the case of R^n, our proof is also based on the Bernstein technique via the maximum principle, hence appears to be simpler and more constructive. However, to avoid possible singularity at the origin, we develop a Bernstein technique in one-dimensional space, which is capable of handling equations in punctured spaces.
ER -