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

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

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