Given any \(R>0\) , we study non-negative, non-trivial, \(C^4\) -solutions to the fourth order Hardy–Hénon equation \(\begin{aligned} \Delta ^2 u = |x|^\sigma u^p \quad \text {in } B_R \setminus \{0\} \subset {{\,\mathrm{{\textbf{R}}}\,}}^n \end{aligned}\) with \(n \ge 2\) , \(p>1\) , and \(\sigma \in {{\,\mathrm{{\textbf{R}}}\,}}\) . While there are many works devoted to the case \(\sigma > -4\) , there are very few works in the scenario \(\sigma \le -4\) . We show in this work that any non-negative, non-trivial, \(C^4\) -solution u to the equation with \(\sigma \le -4\) enjoys a local radial sub harmonic property in the sense that \(\begin{aligned} \int _{\partial B_r} \Delta u d\sigma > 0 \quad \text {for { r} near 0}. \end{aligned}\) In many cases, we are able to show that such a local property is actually global in the sense that the above inequality holds for any \(r \in (0, R)\) . Our argument further reveals the role of the inequality \(n-4-(4+\sigma )/(p-1) \ge 0\) in the analysis of the equation.
