  Combining Carleson-Chang's result 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,
extended by the author to high dimensional case and Riemannian surface case,
generalized by C. Tintarev to wider cases including singular form and by M. de
Souza and J. M. do \'O to the whole Euclidean space R^2. 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, M. Struwe, M.
Flucher, K. Lin, and Adimurthi-Druet, our previous ones, and part of C.
Tintarev.
