A sharp Trudinger-Moser type inequality involving Ln norm in the entire space [formula omitted

Abstract

Let W1,n(Rn) be the standard Sobolev space and ‖⋅‖n be the Ln norm on Rn. We establish a sharp form of the following Trudinger-Moser inequality involving the Ln normsup‖u‖W1,n(Rn)=1∫RnΦ(αn|u|nn−1(1+α‖u‖nn)1n−1)dx<+∞ for any 0≤α<1, where Φ(t)=et−∑n−2j=0tjj!, αn=nωn−11n−1 and ωn−1 is the n−1 dimensional surface measure of the unit ball in Rn. We also show that the above supremum is infinity for all α≥1. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when α>0 is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals.Our results sharpen the recent work [19] in which they show that the above inequality holds in a weaker form when Φ(t) is replaced by a strictly smaller Φ⁎(t)=et−∑n−1j=0tjj! (note that Φ(t)=Φ⁎(t)+tn−1(n−1)!).