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

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 $$J_{\alpha ,\beta }(u)=\frac{1}{2}{\int }_{\Sigma }(|{\nabla }_{g}u{|}^2-\alpha u^2)d{v}_g-\beta \log {\int }_{\Sigma }h{e}^{u}d{v}_g$$ on a function space $\mathcal{H}=\left\{u\in W^{1,2}(\Sigma ):{\int }_{\Sigma }ud{v}_g=0\right\}$. 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 $\alpha \ge {\lambda }_1(\Sigma )$, 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.

23 pages. Accepted by Sci China Math