A generalized Trudinger-Moser inequality on a compact Riemannian surface

Let (Σ,g) be a compact Riemannian surface. Let ψ, h be two smooth functions on Σ with ∫Σψdvg≠0 and h≥0, h≢0. In this paper, using a method of blow-up analysis, we prove that the functional (1)Jψ,h(u)=12∫Σ|∇gu|2dvg+8π1∫Σψdvg∫Σψudvg−8πlog∫Σheudvgis bounded from below in W1,2(Σ,g). Moreover, we obtain a sufficient condition under which Jψ,h attains its infimum in W1,2(Σ,g). These results generalize the main results in Ding-Jost-Li-Wang (1997) and Yang-Zhu (2017).