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

TL;DR

This paper studies the existence and nonexistence of nontrivial nonnegative solutions to fractional p-Laplace inequalities, using analysis of fundamental solutions to understand solution behavior in fractional Sobolev spaces.

Contribution

It provides new results on the existence and nonexistence of solutions to fractional p-Laplace inequalities, employing a novel analysis of fundamental solutions.

Findings

Characterization of conditions for solution existence

Identification of nonexistence regimes based on parameters

Development of analytical techniques for fractional p-Laplace operators

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 $q\in(0,\infty)$. The approach taken in this paper is mainly based on some delicate analysis of the fundamental solutions to the fractional $p$-Laplace operator $(-\Delta)^s_p$.