On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form

In this article, we establish the weighted Trudinger-Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger-Moser inequality and the Caffarelli-Kohn-Nirenberg inequality in the asymptotic sense. (C) 2013 Elsevier Masson SAS. All rights reserved.