Liouville results for supersolutions of fractional $p$-Laplacian equations with gradient nonlinearities

Abstract

We prove that any nonnegative viscosity solution of the inequality $$(-{\Delta }_p{)}^{s}u(x)\ge u^t|\nabla u{|}^m\text{ in }{\mathbb{R}}^N,N\ge 2,$$ must be constant. This result holds for parameters $p\in (1,\infty ),s\in (0,1)$, $t,m\ge 0$, satisfying $$t(N-sp)+m(N-(sp-p+1))<N(p-1),$$ with the additional condition that either $m\le p-1$ if $p-1<sp$, or $m<sp$ if $p-1\ge sp$.