Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

TL;DR

This paper investigates the existence of positive solutions to a class of quasilinear elliptic inequalities involving Hardy potentials and nonlocal Riesz potential terms, providing necessary and sufficient conditions for solutions in unbounded domains.

Contribution

It establishes the precise conditions under which positive solutions exist for a broad class of nonlocal elliptic inequalities with Hardy potentials.

Findings

Derived necessary and sufficient conditions for solution existence

Characterized solution behavior based on parameters p, q, μ, α, m

Extended understanding of nonlocal elliptic inequalities with Hardy potentials

Abstract

We study the quasilinear elliptic inequality $$ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, \mu \in \mathbb{R}$, $m>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.