Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two

Combining Carleson–Chang's result [9] with blow-up analysis, we prove existence of extremal functions for certain Trudinger–Moser inequalities in dimension two. This kind of inequality was originally proposed by Adimurthi and O. Druet [1], extended by the author to high dimensional case and Riemannian surface case [40] and [41], generalized by C. Tintarev to wider cases including singular form [36] and by M. de Souza and J.M. do Ó [14] to the whole Euclidean space R2R2. In addition to the Euclidean case, we also consider the Riemannian surface case. The results in the current paper complement that of L. Carleson and A. Chang [9], M. Struwe [35], M. Flucher [16], K. Lin [19], and Adimurthi and O. Druet [1], our previous ones [26] and [41], and part of C. Tintarev [36].