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.

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