  Inspired by a recent pointwise differential inequality for positive bounded
solutions of the fourth-order H\'enon equation $\Delta^2 u = |x|^a u^p$ in
${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly,
Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive
constants $\alpha$ and $\beta$ we establish the following pointwise inequality
  \[
  \Delta u \geqslant \alpha u^{-\frac{q-1}2} + \beta u^{-1} |\nabla u|^2
  \] in ${\mathbb R}^n$ with $n \geqslant 3$ for positive $C^4$-solutions of
the fourth-order equation
  \[
  \Delta^2u=-u^{-q} \quad \text{ in } \mathbb R^n
  \] where $q > 1$. Next, we prove a comparison property for Lane--Emden system
with exponents of mixed sign. Finally, we give an analogue result for parabolic
models by establishing a comparison property for parabolic system of
Lane--Emden type. To obtain all these results, a new argument of maximum
principle is introduced, which allows us to deal with solutions with high
growth at infinity. We expect to see more applications of this new method to
other problems in different contexts.
