Blow-up analysis on line bundles over a compact Riemann surface with smooth boundary

Using the method of blow-up analysis, we derive two Trudinger-Moser inequalities on a smooth line bundle over a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. Based on the earlier work of Li et al. (Calc Var Partial Differ Equ 28:59–83, 2007), the present paper generalizes Yang’s results (Pacific J Math 227:177–200, 2006 and Internat J Math 17:313–330, 2006). Moreover, we give the existence and uniform estimates of the Green section, which is very important in the blow-up analysis. Although much progress has been made with respect to the existence of extremals for Trudinger-Moser inequalities on Euclidean spaces and Riemann manifolds, there are only a little work on line bundle.