We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_\alpha$ stands for the Riesz potential of order $\alpha\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $\alpha$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.

中文翻译：

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拟线性椭圆不等式和非局部项系统的正解

我们研究了拟线性椭圆不等式 $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\ alpha\ast u^p)u^q$ in $\Omega$，其中 $\Omega\subset \mathbb{R}^N, N\geq 1,$ 是开集。这里 $I_\alpha$ 代表阶 $\alpha\in (0, N)$、$p>0$ 和 $q\in \mathbb{R}$ 的 Riesz 势。对于一大类算子 $L_\mathcal{A}$（包括 $m$-Laplace 和 $m$-mean 曲率算子），我们获得指数 $p,q$ 和 $\alpha$ 的最佳范围存在哪些正解。我们的方法然后扩展到不等式的拟线性椭圆系统。