A weighted form of Moser–Trudinger inequality on Riemannian surface Original Research Article

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