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

Let Omega be a bounded smooth domain in R-n, W-1,n(Omega) be the Sobolev space on Omega, and lambda(Omega) = inf{parallel to del(u)parallel to n n : integral(Omega) udx = 0,parallel to u parallel to(n) = 1} be the first nonzero Neumann eigenvalue of the n-Laplace operator -Delta(n) on Omega. For 0 <= a < lambda(Omega), let us define. parallel to u.parallel to(n)(1), alpha =parallel to del u parallel to(n) (n) - alpha parallel to u parallel to(n) (n). We prove, in this paper, the following improved MoserTrudinger inequality on functions with mean value zero on Omega, u is an element of W-1,W- n(Omega), integral(sup)(Omega)udx= 0,parallel to u parallel to 1,alpha= 1 integral(Omega) e(beta)n| u| n/n-1 dx < infinity, where beta(n) = n(omega(n-1)/2) 1/(n-1), and omega(n-1) denotes the surface area of unit sphere in R-n. We also show that this supremum is attained by some function u*is an element of W-1,W- n(Omega) such that integral(Omega) u* dx = 0 and parallel to u*parallel to(1, alpha) = 1. This generalizes a result of Ngo and Nguyen (0000) in dimension two and a result of Yang (2007) for alpha = 0, and improves a result of Cianchi (2005). (C) 2017 Elsevier Ltd. All rights reserved.