Abstract
We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality
. Here Ω is a bounded domain of ℝN (N ≥ 2), s ∊ (0, 1), sp = N, \(\widetilde{W}_0^{s,p}({\rm{\Omega }})\) is a Sobolev-Slobodeckij space, and \({[ \cdot ]_{{W^{s,p}}({\mathbb{R}^N})}}\) is the associated Gagliardo seminorm. We exhibit an explicit exponent α*s,N > 0, which does not depend on Ω, such that the Moser-Trudinger inequality does not hold true for α ∊ (α*s,N, +∞).
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Parini, E., Ruf, B. On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces. JAMA 138, 281–300 (2019). https://doi.org/10.1007/s11854-019-0029-3
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DOI: https://doi.org/10.1007/s11854-019-0029-3