Revisiting Guo-Huang-Wang-Wei pointwise estimates for radial solutions to the fourth-order H\'enon equation in the punctured space

Abstract

We study pointwise estimates for positive, $C^4$-solution $u$ to the fourth-order H\'enon equation \[ \Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n \setminus \{0\} \] with $n \geq 5$, $p>1$, and $\sigma>-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 \[ -\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac {n(n-4)}{n^2-4}} u^\frac{n}{n-4} \quad \text{in } \mathbf R^n \setminus \{0\} \] for radial solutions to the critical, fourth-order Lane-Emden equation, namely $\sigma = 0$ and $p=(n+4)/(n-4)$. Inspired by this interesting result, we establish a similar pointwise inequality \[ -\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac \sigma 2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n \setminus \{0\} \] for radial solutions to the fourth-order H\'enon equation in $\mathbf R^n \setminus \{0\}$ for any $\sigma \geq 0$ and $p \geq (n+4 + \sigma)/(n-4)$. We also show that the constant $2/(n-4)$ is sharp and that the thresholds $\sigma \geq 0$ and $p \geq (n+4 + \sigma)/(n-4)$ are necessary. As in our previous study on the case of $\mathbf R^n$, 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.