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.

Classification: 46E35, 35J20

Keywords: Weighted Trudinger–Moser inequality, Existence of maximizer, Caffarelli–Kohn–Nirenberg inequality with asymptotics

@article{AIHPC_2014__31_2_297_0, author = {Ishiwata, Michinori and Nakamura, Makoto and Wadade, Hidemitsu}, title = {On the sharp constant for the weighted Trudinger--Moser type inequality of the scaling invariant form}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {31}, number = {2}, year = {2014}, pages = {297-314}, doi = {10.1016/j.anihpc.2013.03.004}, zbl = {1311.46034}, mrnumber = {3181671}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2014__31_2_297_0} }

Ishiwata, Michinori; Nakamura, Makoto; Wadade, Hidemitsu. On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 297-314. doi : 10.1016/j.anihpc.2013.03.004. http://www.numdam.org/item/AIHPC_2014__31_2_297_0/

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