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

The Trudinger-Moser inequality states that for functions u is an element of H-0(1,n)(Omega) (Omega subset of R-n a bounded domain) with integral(Omega) vertical bar del u vertical bar(n) dx <= 1, one has integral(Omega) (e(alpha n vertical bar u vertical bar n/(n-1)) - 1) dx <= c vertical bar Omega vertical bar, with c independent of u. Recently, the second author has shown that for n = 2 the bound c vertical bar Omega vertical bar may be replaced by a uniform constant d independent of Omega if the Dirichlet norm is replaced by the Sobolev norm, i.e., requiring integral(Omega) (vertical bar del u vertical bar(n) + vertical bar u vertical bar(n)) dx <= 1. We extend here this result to arbitrary dimensions n > 2. Also, we prove that for Omega = R-n the supremum of integral(Rn) e(alpha n vertical bar u vertical bar n/(n-1)) - 1) dx over all such functions is attained. The proof is based on a blow-up procedure.