Moser functions and fractional Moser-Trudinger type inequalities

We improve the sharpness of some fractional Moser-Trudinger type inequalities, particularly those studied by Lam-Lu and Martinazzi. As an application, improving upon works of Adimurthi and Lakkis, we prove the existence of weak solutions to the problem $(-Δ)^\frac{n}{2}u=λue^{bu^2} \,\text{ in }Ω,\, 0<λ<λ_1,\,b>0,$ with Dirichlet boundary condition, for any domain $Ω$ in $\mathbb{R}^n$ with finite measure. Here $λ_1$ is the first eigenvalue of $(-Δ)^\frac n2$ on $Ω$.

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