A sharp inequality of Trudinger-Moser type and extremal functions in H1,n(Rn)

We prove a sharp form of the Trudinger-Moser inequality for the Sobolev space H1,n(Rn). The sharpness comes from the introduction of an extra factor ‖u‖nn in the classical Trudinger-Moser inequality. Letℓ(α):=sup{u∈H1,n(Rn):‖u‖1,n=1}⁡∫RnΦ∘να(u)dx, where Φ(t):=et−∑i=0n−1tii! and να(u):=βn(1+α‖u‖nn)1/(n−1)|u|n/(n−1). The main results read: (1) for 0≤α<1 we have ℓ(α)<∞, (2) for α>1, ℓ(α)=∞ and (3) moreover, we prove that for 0≤α<1, an extremal function for ℓ(α) exists.