Isolated Singularities of Nonlinear Polyharmonic Inequalities

TL;DR

This paper investigates the behavior of solutions to nonlinear polyharmonic inequalities near isolated singularities, aiming to identify conditions on the nonlinearity for which solutions have controlled growth and to determine the optimal bounds.

Contribution

It provides new results characterizing the existence and optimal growth bounds of solutions to nonlinear polyharmonic inequalities near singularities.

Findings

Characterization of functions f for which solutions have bounded growth.

Identification of optimal growth bounds near singularities.

Conditions ensuring the existence of solutions with prescribed asymptotic behavior.

Abstract

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