Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms

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.