  We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are
integers.
  Question. For which continuous functions $f\colon [0,\infty)\to [0,\infty)$
does there exist a continuous function $\varphi\colon (0,1)\to (0,\infty)$ such
that every $C^{2m}$ nonnegative solution $u(x)$ of
  $ 0 \le -\Delta^m u\le f(u)\quad$ in $B_2(0)\setminus\{0\}\subset {\bf R}^n $
  satisfies
  $u(x) = O(\varphi(|x|))\quad \text{as } x\to 0 $
  and what is the optimal such $\varphi$ when one exists?
