A weighted form of Moser–Trudinger inequality on Riemannian surface

Let (Σ,g)(Σ,g) be a compact Riemannian surface without boundary. In this paper we shall use blowing up analysis to prove that sup∫Σ|∇u|2fdVg=1,∫ΣudVg=0∫Σe4πfu2dVg<+∞. Furthermore we show that there exists an extremal function for the above inequality. In other words, we show that sup∫Σ|∇u|2fdVg=1,∫ΣudVg=0∫Σe4πfu2dVg is attained.