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

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