Pointwise Bounds and Blow-up for Nonlinear Polyharmonic Inequalities

TL;DR

This paper investigates the bounds and blow-up behavior of solutions to nonlinear polyharmonic inequalities, identifying conditions on the nonlinearity for solutions to be controlled near singularities.

Contribution

It characterizes the existence of optimal pointwise bounds for solutions of polyharmonic inequalities based on the growth of the nonlinear function.

Findings

Established criteria for the existence of bounds near singularities.

Derived optimal bounds for solutions depending on the nonlinearity.

Provided insights into blow-up phenomena for higher-order elliptic inequalities.

Abstract

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. Question. For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ 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?