Isolated Singularities of Nonlinear Polyharmonic Inequalities

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf 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 \text{in}\quad B_2(0)\backslash\{0\}\subset {\bb R}^n satisfies u(x) = O(ϕ(|x|))\quad \text{as}\quad x\to 0 and what is the optimal such $ϕ$ when one exists?

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