On the sub poly-harmonic property for solutions of (-Δ)^p u <0 in R^n

In this note, we mainly study the relation between the sign of $(-\Delta)^p u$ and $(-\Delta)^{p-i} u$ in $\mathbb R^n$ with $p \geqslant 2$ and $n \geqslant 2$ for $1 \leqslant i \leqslant p-1$. Given the differential inequality $(-\Delta)^p u < 0$, first we provide several sufficient conditions so that $(-\Delta)^{p-1} u < 0$ holds. Then we provide conditions such that $(-\Delta)^i u < 0$ for all $i=1,2,...,p-1$ which is known as the sub poly-harmonic property for $u$. In the last part of the note, we revisit the super poly-harmonic property for solutions of $(-\Delta)^p u = e^{2pu}$ and $(-\Delta)^p u = u^q$ with $q>0$ in $\mathbb R^n$.