A sharp Trudinger-Moser type inequality for unbounded domains in Rⁿ

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The Trudinger-Moser inequality states that for functions u ∈ H₀¹,ⁿ(Ω) (Ω⊂ Rⁿ a bounded domain) with ∫Ω|∇ u|ⁿdx ≤ 1 one has ∫Ω(eαₙ|u| nn-1-1)dx ≤ c |Ω|, with c independent of u. Recently, the second author has shown that for n = 2 the bound c |Ω| may be replaced by a uniform constant d independent of Ω if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring ∫Ω(|∇ u|ⁿ + |u|ⁿ)dx ≤ 1. We extend here this result to arbitrary dimensions n > 2. Also, we prove that for Ω= Rⁿ the supremum of ∫ᵣₙ (eαₙ|u| nn-1-1)dx over all such functions is attained. The proof is based on a blow-up procedure.