Nonlinear Inequalities with Double Riesz Potentials

TL;DR

This paper analyzes the existence, nonexistence, and decay properties of nonnegative solutions to a nonlinear integral inequality involving double Riesz potentials, establishing optimal parameter ranges using a nonlocal positivity principle.

Contribution

It introduces a nonlocal positivity principle to determine optimal parameter ranges for solutions of a nonlinear Riesz potential inequality, including decay behavior.

Findings

Derived optimal parameter ranges for existence and nonexistence of solutions.

Established decay rates at infinity for solutions.

Provided a framework for analyzing nonlinear inequalities with Riesz potentials.

Abstract

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote the Riesz potentials of order $\alpha$ and $\beta$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $\alpha$, $\beta$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.