Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order H\'enon 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 \[-\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac \sigma 2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to the the fourth-order H\'enon equation \[\Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n\] with $n \geq 5$, $\sigma \geq 0 $ but implicitly near $0$, and $p>(n+4+2\sigma)/(n-4)$. 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 $p = (n+4+2\sigma)/(n-4)$, but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any $\sigma \geq 0$. This estimate leads to an interesting consequence of the existence of conformal metric of $\mathbf R^n$ 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.