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Abstract. Let (M, g) be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over (M, g). We also derive an upper bound of such inequalities under the assumption that blowing up occur.

Combining Carleson-Chang's result [9] with blow-up analysis, we prove existence of extremal functions for certain Trudinger-Moser inequalities in dimension two.

Abstract. In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved.

Let G = (V, E) be a finite graph and ∆ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan-Warner equation ∆u = c − he u has a solution on V, where c is a constant, and h : V → R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan-Warner (Ann. Math., 1974).