  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
$\Omega$.
