Isolated Singularities of Polyharmonic Inequalities

TL;DR

This paper investigates conditions under which nonnegative solutions to polyharmonic inequalities near singular points are bounded, providing a comprehensive characterization based on dimension and order of the operator.

Contribution

It establishes necessary and sufficient conditions for pointwise bounds of solutions to polyharmonic inequalities near isolated singularities.

Findings

Derived criteria for boundedness of solutions near singularities

Identified critical dimensions and orders for solution behavior

Enhanced understanding of polyharmonic inequality solutions in punctured domains

Abstract

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.