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

This note shows that under $(p,alpha, N)in (1,infty)times(0,2)timesmathbb Z_+$ the fractional order differential inequality $$ (dagger)quad u^p le (-Delta)^{frac{alpha}{2}} uquadhbox{in}quadmathbb R^{N} $$ has the property that if $Nlealpha$ 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 $ple N/(N-alpha)$, thereby innovatively generalizing Gidas-Sprucks result for $u^p+Delta ule 0$ in $R^N$ discovered in cite{GS}.