Fractional optimal maximization problem and the unstable fractional obstacle problem

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 $\alpha>0$ $$(-\Delta)^s u=\chi_{\{u>\alpha\}}.$$