  We consider an optimal rearrangement maximization problem involving the
fractional Laplace operator $(-\Delta)^s$, $0<s<1$, and the Gagliardo-Nirenberg
seminorm $[u]_s$. We prove the existence of a maximizer, analyze its properties
and show that it satisfies the unstable fractional obstacle problem equation
for some $\alpha>0$ $$(-\Delta)^s u=\chi_{\{u>\alpha\}}.$$
