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
\[
(−∆)^{n/2} u = \lambda u e^{bu^2} in \Omega, 0 < \lambda < \lambda_1, b > 0,
\]
with Dirichlet boundary condition, for any domain $\Omega$ in $\mathbb{R}$ with finite measure. Here $\lambda_1$ is the first eigenvalue of $(−∆)^{n/2}$ on $\Omega$.