TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 7, July 1996 EXTREMAL FUNCTIONS FOR MOSER’S INEQUALITY KAI-CHING LIN Abstract. Let Ω be a bounded smooth domain in R n , and u(x)a C 1 function with compact support in Ω. Moser’s inequality states that there is a constant co, depending only on the dimension n, such that 1 |Ω|  Ω e nω 1 n-1 n-1 u n n-1 dx ≤ co, where |Ω| is the Lebesgue measure of Ω, and ω n−1 the surface area of the unit ball in R n . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the sup{ 1 |Ω|  Ω e nω 1 n-1 n-1 u n n-1 dx : u ∈ W 1,n o , ‖∇u‖n ≤ 1} is attained. Earlier results include Carleson-Chang (1986, Ω is a ball in any dimension) and Flucher (1992, Ω is any domain in 2-dimensions). 1. Introduction Let Ω be a bounded smooth domain in R n , and u(x)a C 1 function supported in Ω with ‖∇u‖ q <n. Sobolev’s Imbedding Theorem says that if 1 ≤ q<n, then ‖u‖ p ≤ C(n, q), (1) where 1 p = 1 q − 1 n , and C(n, q) is a constant independent of the function u, as well as the domain Ω. The imbedding is no longer valid when q = n. Indeed, there are unbounded functions whose gradients are in L n . However, Trudinger [14] in 1967 proved that if ‖∇u‖ n ≤ 1, then u is in an exponential class. More precisely, the integral  Ω e βou n n-1 dx, is uniformly bounded, for some positive β 0 depending only on dimension. Moser [12] in 1971 then found the best exponent β 0 . He showed if ‖∇u‖ n ≤ 1, then 1 |Ω|  Ω e nω 1 n-1 n-1 u n n-1 dx ≤ c 0 , (2) where c 0 is a constant depending only on n.(ω n−1 is the surface area of the unit ball in R n .) The aim of this paper is to prove the following: Received by the editors January 25, 1995 and, in revised form, May 30, 1995. 1991 Mathematics Subject Classification. Primary 49J10. c 1996 American Mathematical Society 2663 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use