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

Abstract

Let $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^n$, $W^{1,n}(\Omega )$ be the Sobolev space on $\Omega$, and $\lambda (\Omega )=\inf \{\Vert \nabla u{\Vert }_n^n:{\int }_{\Omega }udx=0,\Vert u{\Vert }_n=1\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-{\Delta }_n$ on $\Omega$. For $0\le \alpha <\lambda (\Omega )$, let us define $\Vert u{\Vert }_{1,\alpha }^n=\Vert \nabla u{\Vert }_n^n-\alpha \Vert u{\Vert }_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\Omega$, \[ \sup_{u\in W^{1,n}(Ω), \int_Ωu dx =0, \|u\|_{1,α} =1} \int_Ω e^{β_n |u|^{\frac n{n-1}}} dx < \infty, \] 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 $u^{\ast }\in W^{1,n}(\Omega )$ such that ${\int }_{\Omega }{u}^{\ast }dx=0$ and $\Vert u^{\ast }{\Vert }_{1,\alpha }=1$. 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}.

arXiv comment24 pages, to appear in Nonlinear Analysis