Revisiting Fazly-Wei-Xu pointwise estimates for the fourth-order Hénon equation via Bernstein's technique

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 \[-Δu \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac σ2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounde