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 $\int_Σψdv_g \neq 0$ and $h\geq0$, $h\not\equiv0$. In this paper, using a method of blowup analysis, we prove that the functional \begin{align}\label{functional_J} J^{ψ,h}(u)=\frac{1}{2}\int _Σ|\nabla_g u|^2dv_g + 8π\frac{1}{\int_Σψdv_g}\int_Σψudv_g-8π\log\int _Σhe^{u}dv_g \end{align} is bounded from below in $W^{1,2}(Σ,g)$. Moreover, we obtain a sufficient condition under which $J^{ψ, h}$ attains its infimum in $W^{1,2}(Σ,g)$. These results generalize the main results in \cite{DJLW97} and \cite{YZ2016}.

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