Abstract 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 in W s , p ( ℝ N ) {W^{s,p}(\mathbb{R}^{N})} . Define ∥ u ∥ 1 , τ = ( [ u ] W s , p ( ℝ N ) p + τ ∥ u ∥ p p ) 1 p for any τ > 0 . \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds sup u ∈ W s , p ( ℝ N ) , ∥ u ∥ 1 , τ ≤ 1 ∫ ℝ N Φ N , s ( α | u | N N - s ) < + ∞ , \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where s ∈ ( 0 , 1 ) {s\in(0,1)} , s p = N {sp=N} , α ∈ [ 0 , α * ) {\alpha\in[0,\alpha_{*})} and Φ N , s ( t ) = e t - ∑ i = 0 j p - 2 t j j ! . \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: ( - Δ ) p s u ( x ) + V ( x ) | u ( x ) | p - 2 u ( x ) = f ( x , u ) + ε h ( x ) in ℝ N , (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where V ( x ) {V(x)} has a positive lower bound, f ( x , t ) {f(x,t)} behaves like e α | t | N / ( N - s ) {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , h ∈ ( W s , p ( ℝ N ) ) * {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and ε > 0 {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.