A sharp Trudinger-Moser type inequality for unbounded domains in R^n

The Trudinger-Moser inequality states that for functions u in the Sobolev space H^1,n over a bounded domain with a certain exponential integral is bounded by a constant depending on the domain, but not on u. Recently, the second author has shown that for n = 2 the bound the constant is independent of the domain if the Dirichlet norm is replaced by the full Sobolev norm, We extend here this result to arbitrary dimensions n > 2. Also, we prove that on all of R^n the corresponding supremum is attained. The proof is based on a blow-up procedure.