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

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

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