Improved Moser--Trudinger inequality for functions with mean value zero in $\mathbb R^n$ and its extremal functions

Abstract

Let $Ω$ be a bounded smooth domain in $\mathbb R^n$, $W^{1,n}(Ω)$ be the Sobolev space on $Ω$, and $λ(Ω) = \inf\{\|\nabla u\|_n^n: \int_Ωu dx =0, \|u\|_n =1\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-Δ_n$ on $Ω$. For $0 \leq α< λ(Ω)$, let us define $\|u\|_{1,α}^n =\|\nabla u\|_n^n -α\|u\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $Ω$, \[ \sup_{u\in W^{1,n}(Ω), \int_Ωu dx =0, \|u\|_{1,α} =1} \int_Ω e^{β_n |u|^{\frac n{n-1}}} dx < \infty, \] where $β_n = n (ω_{n-1}/2)^{1/(n-1)}$, and $ω_{n-1}$ denotes the surface area of unit sphere in $\mathbb R^n$. We also show that this supremum is attained by some function $u^*\in W^{1,n}(Ω)$ such that $\int_Ωu^* dx =0$ and $\|u^*\|_{1,α} =1$. This generalizes a result of Ngo and Nguyen \cite{NN17} in dimension two and a result of Yang \cite{Yang07} for $α=0$, and improves a result of Cianchi \cite{Cianchi05}.