Improved Moser-Trudinger inequality for functions with mean value zero in Rn and its extremal functions

Let Ω be a bounded smooth domain in Rn, W1,n(Ω) be the Sobolev space on Ω, and λ(Ω)=inf{‖∇u‖nn:∫Ωudx=0,‖u‖n=1} be the first nonzero Neumann eigenvalue of the n−Laplace operator −Δn on Ω. For 0≤α<λ(Ω), let us define ‖u‖1,αn=‖∇u‖nn−α‖u‖nn. We prove, in this paper, the following improved Moser-Trudinger inequality on functions with mean value zero on Ω, supu∈W1,n(Ω),∫Ωudx=0,‖u‖1,α=1∫Ωeβn|u|nn−1dx<∞,where βn=n(ωn−1∕2)1∕(n−1), and ωn−1 denotes the surface area of unit sphere in Rn. We also show that this supremum is attained by some function u∗∈W1,n(Ω) such that ∫Ωu∗dx=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).