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

Abstract: For the nonlocal quasilinear fractional $p$-Laplace operator $(-\Delta)^s_p$ 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 $W_{\rm loc}^{s,p}(\mathbb R^n)$ that satisfies the inequality $(-\Delta)^s_p u\ge u^q$ weakly in $\mathbb R^n$, where $q\in(0,\infty)$. In addition, nonexistence of nontrivial nonnegative weak solutions in the global fractional Sobolev space $W^{s,p}(\mathbb R^n)$ to the fractional $p$-Laplace equation $(-\Delta)^s_p u= u^q$ are also investigated. The approach taken in this paper is mainly based on some delicate analysis of the fundamental solutions to the fractional $p$-Laplace operator $(-\Delta)^s_p$.