Pointwise Bounds and Blow-up for Nonlinear Polyharmonic Inequalities

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 $φ\colon (0,1)\to (0,\infty)$ such that every $C^{2m}$ nonnegative solution $u(x)$ of $0 \le -Δ^m u\le f(u)\quad$ in $B_2(0)\setminus\{0\}\subset {\bf R}^n$ satisfies $u(x) = O(φ(|x|))\quad \text{as } x\to 0$ and what is the optimal such $φ$ when one exists?

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