Moser–Trudinger inequality for functions with mean value zero

Let ΩΩ be a bounded smooth domain in Rn(n≥2). This paper deals with the Moser–Trudinger inequality for functions with mean value zero. Using blowing up analysis, the author proves that sup{∫Ωeα|u|nn−1dx:u∈H1,n(Ω),∫Ω|∇u|ndx=1,∫Ωudx=0} is attained for any α≤αnα≤αn, and the supremum is infinity for any α>αnα>αn, where αn=n(ωn−1/2)1/(n−1)αn=n(ωn−1/2)1/(n−1), and ωn−1ωn−1 is the area of the unit sphere in RnRn.