A generalized Trudinger-Moser inequality on a compact Riemannian surface

Let $(\mathrm{\Sigma },g)$ be a compact Riemannian surface. Let $\psi$, $h$ be two smooth functions on $\mathrm{\Sigma }$ with ${\int }_{\mathrm{\Sigma }}\psi d{v}_g\mathrm{\ne }0$ and $h\ge 0$, $h\equiv ̸0$. In this paper, using a method of blowup analysis, we prove that the functional \begin{align}\label{functional_J} J^{\psi,h}(u)=\frac{1}{2}\int {\Sigma}|\nabla_g u|^2dv_g + 8\pi\frac{1}{\int\Sigma \psi dv_g}\int_\Sigma \psi udv_g-8\pi\log\int _{\Sigma}he^{u}dv_g \end{align} is bounded from below in $W^{1,2}(\mathrm{\Sigma },g)$. Moreover, we obtain a sufficient condition under which $J^{\psi ,h}$ attains its infimum in $W^{1,2}(\mathrm{\Sigma },g)$. These results generalize the main results in \cite{DJLW97} and \cite{YZ2016}.