A Weak Trudinger–Moser Inequality with a Singular Weight on a Compact Riemannian Surface

Abstract

Let $(\Sigma ,g)$ be a compact Riemannian surface, $p_j\in \Sigma$, $\beta _{j}>-1$, for $j=1,\cdots ,m$. Denote $\beta =\min \{0,\beta _1,\cdots ,\beta _{m}\}$. Let $H\in C^0(\Sigma )$ be a positive function and $h(x)=H(x)\left( d_g(x,p_j)\right) ^{2\beta _j}$, where $d_g(x,p_j)$ denotes the geodesic distance between x and $p_j$ for each $j=1,\cdots ,m$. In this paper, using a method of blow-up analysis, we prove that the functional

$$\begin{aligned} J(u)=\frac{1}{2}\int _{\Sigma }|\nabla _g u|^2dv_g + 8\pi (1+\beta )\frac{1}{Vol_g(\Sigma )} \int _{\Sigma }udv_g-8\pi (1+\beta ) \log \int _{\Sigma }he^{u}dv_g \end{aligned}$$

is bounded from below on the Sobolev space $W^{1,2}(\Sigma ,g)$.