Nontrivial nonnegative weak solutions to fractional $p$-Laplace inequalities

For the nonlocal quasilinear fractional $p$-Laplace operator $(-\mathrm{\Delta }{)}_p^s$ with $s\in (0,1)$ and $p\in (1,\mathrm{\infty })$, we investigate the nonexistence and existence of nontrivial nonnegative solutions $u$ in the local fractional Sobolev space $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{s,p}({\mathbb{R}}^n)$ that satisfies the inequality $(-\mathrm{\Delta }{)}_p^{s}u\ge u^q$ weakly in ${\mathbb{R}}^n$, where $q\in (0,\mathrm{\infty })$. The approach taken in this paper is mainly based on some delicate analysis of the fundamental solutions to the fractional $p$-Laplace operator $(-\mathrm{\Delta }{)}_p^s$.