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In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality inW   s  ,  p    ⁢   (   ℝ  N   ){W^{s,p}(\mathbb{R}^{N})}   . Define

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation