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

Abstract

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

v2;42 pp