A sharp Trudinger-Moser type inequality involving Lⁿ norm in the entire space Rⁿ

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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 norm u W¹,ⁿ(R ⁿ)=1∫ᵣⁿΦ(αₙ u ⁿ/ⁿ⁻¹(1+α u ₙⁿ) ¹/ⁿ⁻¹) dx<+∞ in the entire space Rⁿ for any 0≤α<1, where Φ(t) =eᵗ-j=0n-2∑% tʲj!, αₙ=nωₙ₋₁¹/ⁿ⁻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 result sharpens the recent work J. M. do1 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-j=0n-1∑% tʲj!. (Note that Φ(t)=Φ*(t)+tⁿ⁻¹(n-1)!).