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 $(-\Delta)^\frac{n}{2}u=\lambda ue^{bu^2} \,\text{ in }\Omega,\, 0<\lambda<\lambda_1,\,b>0,$ with Dirichlet boundary condition, for any domain $\Omega$ in $\mathbb{R}^n$ with finite measure. Here $\lambda_1$ is the first eigenvalue of $(-\Delta)^\frac n2$ on $\Omega$.