Improved Moser–Trudinger inequality for functions with mean value zero in [formula omitted] and its extremal functions.

Let Ω be a bounded smooth domain in R n , W 1 , n ( Ω ) be the Sobolev space on Ω , and λ ( Ω ) = inf { ‖∇u‖ n n : ∫ Ω u d x = 0 , ‖ u ‖ n = 1 } be the first nonzero Neumann eigenvalue of the n − Laplace operator − Δ n on Ω . For 0 ≤ α < λ ( Ω ) , let us define ‖u‖ 1 , α n = ‖∇u‖ n n − α ‖u‖ n n . We prove, in this paper, the following improved Moser–Trudinger inequality on functions with mean value zero on Ω , sup u ∈ W 1 , n ( Ω ) , ∫ Ω u d x = 0 , ‖ u ‖ 1 , α = 1 ∫ Ω e β n | u | n n − 1 d x < ∞ , where β n = n ( ω n − 1 ∕ 2 ) 1 ∕ ( n − 1 ) , and ω n − 1 denotes the surface area of unit sphere in R n . We also show that this supremum is attained by some function u ∗ ∈ W 1 , n ( Ω ) such that ∫ Ω u ∗ d x = 0 and ‖ u ∗ ‖ 1 , α = 1 . This generalizes a result of Ngo and Nguyen (0000) in dimension two and a result of Yang (2007) for α = 0 , and improves a result of Cianchi (2005). [ABSTRACT FROM AUTHOR]