A sharp form of Moser–Trudinger inequality in high dimension

Let Ω   be a bounded smooth domain in RnRn(n⩾3)(n⩾3). This paper deals with a sharp form of Moser–Trudinger inequality. Letλ1(Ω)=infu∈H01,n(Ω),u≢0‖∇u‖nn/‖u‖nn be the first eigenvalue associated with n-Laplacian. Using blowing up analysis, the author proves thatsupu∈H01,n(Ω),‖∇u‖n=1∫Ωeαn(1+α‖u‖nn)1n−1|u|nn−1dx is finite for any 0⩽α<λ1(Ω)0⩽α<λ1(Ω), and the supremum is infinity for any α⩾λ1(Ω)α⩾λ1(Ω), where αn=nωn−11/(n−1), ωn−1ωn−1 is the surface area of the unit ball in RnRn. Furthermore, the supremum is attained for any 0⩽α<λ1(Ω)0⩽α<λ1(Ω).