A New Proof of Subcritical Trudinger-Moser Inequalities on the Whole Euclidean Space

Abstract

In this note, we give a new proof of subcritical Trudinger-Moser inequality on $R^n$. All the existing proofs on this inequality are based on the rearrangement argument with respect to functions in the Sobolev space $W^{1,n}(R^n)$. Our method avoids this technique and thus can be used in the Riemannian manifold case and in the entire Heisenberg group.