Moser functions and fractional Moser-Trudinger type inequalities

Abstract

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 $(-\hat{I}”{)}^{\frac{n}{2}}u=\hat{I}»u{e}^{b{u}^2}\text{ in }\hat{I}\text{c◯},0<\hat{I}»<\hat{I}{»}_1,b>0,$ with Dirichlet boundary condition, for any domain $\hat{I}\text{c◯}$ in ${\mathbb{R}}^n$ with finite measure. Here $\hat{I}{»}_1$ is the first eigenvalue of $(-\hat{I}”{)}^{\frac{n}{2}}$ on $\hat{I}\text{c◯}$.