Improved Moser--Trudinger inequality for functions with mean value zero in $\mathbb R^n$ and its extremal functions (1708.03028v1)

Abstract: 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}.