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

Abstract

We study the inequality $-\hat{I}”u-\frac{\hat{I}}{¼}|x{|}^{2}u\ge (|x{|}^{-\hat{I}\pm }\ast u^p)u^q$ in an unbounded cone ${\mathcal{C}}_{\hat{I}}{\text{c◯}}^{\ddot{I}}\subset {\mathbb{R}}^N$ ($N\ge 2$) generated by a subdomain $\hat{I}\text{c◯}$ of the unit sphere $S^{N-1}\subset {\mathbb{R}}^N,$ $p,q,\ddot{I}>0$, $\hat{I}¼\in \mathbb{R}$ and $0\le \hat{I}\pm <N$. In the above, $|x{|}^{-\hat{I}\pm }\ast u^p$ denotes the standard convolution operator in the cone ${\mathcal{C}}_{\hat{I}}{\text{c◯}}^{\ddot{I}}$. We discuss the existence and nonexistence of positive solutions in terms of $N,p,q,\hat{I}\pm ,\hat{I}¼$ and $\hat{I}\text{c◯}$. Extensions to systems of inequalities are also investigated.