A sharp Trudinger-Moser Type Inequality for Unbounded Domains and Higher Order Derivatives

Abstract

The Trudinger- Moser inequality states that for functions  of  bounded domain  with  one has  , with    independent of  . It is shown that for    the  bound    may  be replaced by a uniform constant  independent of  if the Drichlet norm is replaced by the Sobolev norm.In this paper the results for  have been showed with a lower bound and gradient estimate.