# Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

## TL;DR

This paper investigates the existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surfaces, using blow-up analysis to determine conditions for the functional's infimum and solution existence.

## Contribution

It provides new criteria for the existence of solutions to Kazdan-Warner equations by analyzing the functional's behavior and employing blow-up analysis techniques.

## Key findings

- If <(), the infimum of the functional is calculated using blow-up analysis.
- A sufficient condition for the existence of solutions to the Kazdan-Warner equation is established.
- The functional is unbounded below when  or >8, indicating no minimizers in these cases.

## Abstract

Let $(\Sigma,g)$ be a compact Riemannian surface without boundary and $\lambda_1(\Sigma)$ be the first eigenvalue of the Laplace-Beltrami operator $\Delta_g$. Let $h$ be a positive smooth function on $\Sigma$. Define a functional $$J_{\alpha,\beta}(u)=\frac{1}{2}\int_\Sigma(|\nabla_gu|^2-\alpha u^2)dv_g-\beta\log\int_\Sigma he^udv_g$$ on a function space $\mathcal{H}=\left\{u\in W^{1,2}(\Sigma): \int_\Sigma udv_g=0\right\}$. If $\alpha<\lambda_1(\Sigma)$ and $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, then we calculate the infimum of $J_{\alpha,8\pi}$ on $\mathcal{H}$ by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Warner equation has a solution. If $\alpha\geq \lambda_1(\Sigma)$, then $\inf_{u\in\mathcal{H}}J_{\alpha,8\pi}(u)=-\infty$. If $\beta>8\pi$, then for any $\alpha\in\mathbb{R}$, there holds $\inf_{u\in\mathcal{H}}J_{\alpha,\beta}(u)=-\infty$. Moreover, we consider the same problem in the case that $\alpha$ is large, where higher order eigenvalues are involved.