On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality \[ \sup \left\{\int_\Omega \exp{\left(\alpha\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in \widetilde{W}^{s,p}_0(\Omega),\,[u]_{W^{s,p}(\mathbb{R}^N)}\leq 1 \right\}< + \infty.\] Here $\Omega$ is a bounded domain of $\mathbb{R}^N$ ($N\geq 2$), $s \in (0,1)$, $sp = N$, $\widetilde{W}^{s,p}_0(\Omega)$ is a Sobolev-Slobodeckij space, and $[\cdot]_{W^{s,p}(\mathbb{R}^N)}$ is the associated Gagliardo seminorm. We exhibit an explicit exponent $\alpha^*_{s,N}>0$, which does not depend on $\Omega$, such that the Moser-Trudinger inequality does not hold true for $\alpha \in (\alpha^*_{s,N},+\infty)$.