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

Ishiwata, Michinori; Nakamura, Makoto; Wadade, Hidemitsu

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 297-314.

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.

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