Elliptic inequalities with nonlinear convolution and Hardy terms in cone-like domains

TL;DR

This paper investigates elliptic inequalities with nonlinear convolution and Hardy potential terms in cone-like domains, establishing conditions for the existence and nonexistence of positive solutions and extending results to systems.

Contribution

It provides new criteria for existence and nonexistence of solutions to complex elliptic inequalities with convolution and Hardy terms in cone domains, including system extensions.

Findings

Derived conditions for positive solution existence.

Identified parameter regimes for nonexistence.

Extended analysis to systems of inequalities.

Abstract

We study the inequality $ -\Delta u - \frac{\mu}{|x|^2} u \geq (|x|^{-\alpha} * u^p)u^q$ in an unbounded cone $\mathcal{C}_\Omega^\rho\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $\Omega$ of the unit sphere $S^{N-1}\subset \mathbb{R}^N,$ $p, q, \rho>0$, $\mu\in \mathbb{R}$ and $0\leq \alpha < N$. In the above, $|x|^{-\alpha} * u^p$ denotes the standard convolution operator in the cone $\mathcal{C}_\Omega^\rho$. We discuss the existence and nonexistence of positive solutions in terms of $N, p, q, \alpha, \mu$ and $\Omega$. Extensions to systems of inequalities are also investigated.