Pointwise Bounds and Blow-up for Nonlinear Polyharmonic Inequalities

Abstract

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

32 pages. This is a much-improved version of "Isolated Singularities of Nonlinear Polyharmonic Inequalities" arXiv:1107.4586