Extremal Functions for a Trudinger-Moser Inequality with a Sign-Changing Weight

Let \((\Sigma ,g)\) $(\mathrm{\Sigma },g)$ be a closed Riemann surface, \(\lambda _1(\Sigma )\) ${\lambda }_1(\mathrm{\Sigma })$ be the first eigenvalue of the Laplace-Beltrami operator. Assume \(h:\Sigma \rightarrow \mathbb {R}\) $h:\mathrm{\Sigma }\to \mathbb{R}$ is some smooth sign-changing function. Using blow-up analysis, we prove that for any \(\alpha <\lambda _1(\Sigma )\) $\alpha <{\lambda }_1(\mathrm{\Sigma })$ , the supremum \(\sup _{\int _\Sigma |\nabla _gu|^2dv_g-\alpha \int _\Sigma u^2dv_g\le 1,\,\int _\Sigma udv_g=0}\int _\Sigma he^{4\pi u^2}dv_g\) $$\underset{{\int }_{\mathrm{\Sigma }}{|{\mathrm{\nabla }}_{g}u|}^{2}d{v}_g-\alpha {\int }_{\mathrm{\Sigma }}{u}^{2}d{v}_g\le 1,{\int }_{\mathrm{\Sigma }}ud{v}_g=0}{sup}{\int }_{\mathrm{\Sigma }}h{e}^{4\pi u^2}d{v}_g$$ is attained by some admissible function \(u_\alpha \) $u_{\alpha }$ . This generalizes earlier results of Yang (J. Differential Equations 2015) and Hou (J. Math. ineq. 2018). Our result resembles existence of solutions to the mean field equations \(\Delta _gu=8\pi \left( \frac{he^u}{\int _\Sigma he^udv_g}-\frac{1}{|\Sigma |}\right) ,\) $${\mathrm{\Delta }}_{g}u=8\pi \left(\frac{h{e}^u}{{\int }_{\mathrm{\Sigma }}h{e}^{u}d{v}_g},-,\frac{1}{|\mathrm{\Sigma }|}\right),$$ where h is a smooth sign-changing function. Such problems were extensively studied by L. Sun and J. Y. Zhu (Cal. Var. 2021; arXiv: 2012.12840).