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 $L_{\mathcal{A}}u=-\mathrm{div}[\mathcal{A}(x,u,\nabla u)]\ge (I_{\alpha }\ast u^p)u^q$ in $\Omega$, where $\Omega \subset {\mathbb{R}}^N,N\ge 1,$ 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.

References in corpus (4)

• A guide to the Choquard equation
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
• Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains
• Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Cited by in corpus (4)

• Quasilinear elliptic inequalities with Hardy potential and nonlocal terms
• Fujita type results for quasilinear parabolic inequalities with nonlocal terms
• Polyharmonic inequalities with nonlocal terms
• Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms