A sharp form of Moser–Trudinger inequality in high dimension

Let Ω be a bounded smooth domain in Rn (n 3). This paper deals with a sharp form of Moser– Trudinger inequality. Let λ1(Ω) = inf u∈H 1,n 0 (Ω),u ≡0 ∇u nn / u nn be the first eigenvalue associated with n-Laplacian. Using blowing up analysis, the author proves that sup u∈H 1,n 0 (Ω), ∇u n=1 Ω eαn(1+α u nn ) 1 n−1 |u| n n−1 dx is finite for any 0 α <λ1(Ω), and the supremum is infinity for any α λ1(Ω), where αn = nω 1/(n−1) n−1 , ωn−1 is the surface area of the unit ball in Rn. Furthermore, the supremum is attained for any 0 α <λ1(Ω). © 2006 Elsevier Inc. All rights reserved.