Polyharmonic inequalities with nonlocal terms

TL;DR

This paper investigates conditions for the existence and non-existence of classical solutions to polyharmonic inequalities involving nonlocal convolution terms, establishing new methods and Liouville type results for scalar and system cases.

Contribution

It introduces novel techniques to analyze polyharmonic inequalities with nonlocal terms, including solutions' poly-superharmonic properties and Liouville theorems, extending to systems.

Findings

Solutions satisfy poly-superharmonic property

Liouville type non-existence results established

Methods applicable to systems of inequalities

Abstract

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $\Psi>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.