Uniqueness of nonnegative weak solution to u^ple(-Delta)^frac{α}{2}u on mathbb R^N

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}.