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

Abstract

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

accepted for publication in Journal d'Analyse Mathématique