Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls

TL;DR

This paper investigates the conditions under which positive singular solutions exist or do not exist for certain second-order elliptic inequalities with measurable coefficients, identifying a critical exponent that determines this behavior.

Contribution

It introduces a critical value p* that delineates existence and non-existence regions and provides optimal conditions for the critical case based on coefficient stabilization.

Findings

Existence of a critical exponent p* separating solution regimes

Conditions for solution existence depend on coefficient stabilization rate

Optimal conditions established for the critical case p=p*

Abstract

We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence region from non-existence. In the critical case $p=p^*$ we show that the existence of a singular solution depends on the rate at which the coefficients stabilize at zero and we provide some optimal conditions in this setting.