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

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.