Uniqueness of nonnegative weak solution to $u^p\le(-\Delta)^\frac{\alpha}{2}u$ on $\mathbb R^N$

TL;DR

This paper establishes the conditions under which nonnegative weak solutions to a fractional differential inequality are unique in \\mathbb{R}^N, generalizing classical results to fractional Laplacians and different dimensions.

Contribution

It provides a comprehensive uniqueness criterion for solutions to a fractional differential inequality, extending classical results to fractional orders and higher dimensions.

Findings

Uniqueness holds if N ≤ α for the inequality.

When N > α, uniqueness occurs if and only if p ≤ N/(N - α).

Generalizes Gidas-Spruck's classical result to fractional Laplacian context.

Abstract

This note shows that under $(p,\alpha, N)\in (1,\infty)\times(0,2)\times\mathbb Z_+$ the fractional order differential inequality $$ (\dagger)\quad u^p \le (-\Delta)^{\frac{\alpha}{2}} u\quad\hbox{in}\quad\mathbb R^{N} $$ has the property that if $N\le\alpha$ then a nonnegative solution to $(\dagger)$ is unique, and if $N>\alpha$ then the uniqueness of a nonnegative weak solution to $(\dagger)$ occurs when and only when $p\le N/(N-\alpha)$, thereby innovatively generalizing Gidas-Spruck's result for $u^p+\Delta u\le 0$ in $\R^N$ discovered in \cite{GS}.