Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique
Abstract
In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate −Δu ≥ 2/(n−4) (|∇ u|2)/u + √(2/(p+1− 8/(n(n−4)))) |x|σ/2 u/p+12 in Rn for any bounded, positive, C4-solution u to the the fourth-order Hénon equation Δ2 u = |x|σup in Rn with n ≥ 5, $σ\geq 0 $ but implicitly near $0$, and p>(n+4+2σ)/(n−4). Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for p = (n+4+2σ)/(n−4), but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any σ≥ 0. This estimate leads to an interesting consequence of the existence of conformal metric of Rn having positive scalar and Q curvatures. We also show that the coefficient 2/(n−4) is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.
Related
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BibTeX
@misc{ngo2026revisiting,
title = {{Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique}},
author = {Quôc Anh Ngô and Trung Nguyen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30625},
eprint = {2608.30625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30625},
note = {CC0},
}
RIS
TY - UNPB
TI - Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique
AU - Quôc Anh Ngô
AU - Trung Nguyen
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30625
UR - https://arxiv.org/abs/2608.30625
AB - In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate −Δu ≥ 2/(n−4) (|∇ u|^2)/u + √(2/(p+1− 8/(n(n−4)))) |x|^σ/2 u^/p+12 in R^n for any bounded, positive, C^4-solution u to the the fourth-order Hénon equation Δ^2 u = |x|^σu^p in R^n with n ≥ 5, $σ\geq 0 $ but implicitly near $0$, and p>(n+4+2σ)/(n−4). Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for p = (n+4+2σ)/(n−4), but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any σ≥ 0. This estimate leads to an interesting consequence of the existence of conformal metric of R^n having positive scalar and Q curvatures. We also show that the coefficient 2/(n−4) is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.
ER -