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)n2u=λuebu2in  Ω,0<λ<λ1,b>0, with Dirichlet boundary condition, for any domain Ω in Rn with finite measure. Here λ1 is the first eigenvalue of (−Δ)n2 on Ω.