Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms

Abstract

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_{\mathcal{A}}u=-\mathrm{div}[\mathcal{A}(x,u,\nabla u)]\ge (I_{\hat{I}}\pm \ast u^p)u^q$ in $\hat{I}\text{c◯}$, where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N,N\ge 1,$ is an open set. Here $I_{\hat{I}}\pm$ stands for the Riesz potential of order $\hat{I}\pm \in (0,N)$, $p>0$ 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 $\hat{I}\pm$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.