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

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