Existence of Solutions to a Class of Kazdan-Warner Equations on Finite Graphs

Let $G=(V,E)$ be a connected finite graph, $h$ be a positive function on $V$ and ${\lambda }_1(V)$ be the first non-zero eigenvalue of $-\mathrm{\Delta }$. For any given finite measure $\mu$ on $V$, define functionals \begin{eqnarray*} J_{ β}(u)&=&\frac{1}{2}\int_{V}|\nabla u|^{2}d μ-β\log\int_{V}he^{u}d μ, J_{ α,β}(u)&=&\frac{1}{2}\int_{V}\left(|\nabla u|^{2}- αu^{2}\right) d μ-β\log\int_{V}he^{u}d μ\end{eqnarray*} on the functional space $$\mathbf{H}=\{u\in {\mathbf{W}}^{1,2}(V)\mid {\int }_{V}u\text{\hspace{0.17em}⁣}d\mu =0\}\mathrm{.}$$ For any $\beta \in \mathbb{R}$, we show that $J_{\beta }(u)$ has a minimizer $u\in \mathbf{H}$, and then, based on variational principle, the Kazdan-Warner equation $$\mathrm{\Delta }u=-\frac{\beta h{e}^u}{{\displaystyle {\int }_{V}h{e}^{u}d\mu }}+\frac{\beta }{\text{Vol}(V)}$$ has a solution in $\mathbf{H}$. If $\alpha <{\lambda }_1(V)$, then for any $\beta \in \mathbb{R},J_{\alpha ,\beta }(u)$ has a minimizer in $\mathbf{H}$, thus the Kazdan-Warner equation $$\mathrm{\Delta }u+\alpha \text{\hspace{0.17em}⁣}u=-\frac{\beta h{e}^u}{{\displaystyle {\int }_{V}h{e}^{u}d\mu }}+\frac{\beta }{\text{Vol}(V)}$$ has a solution in $\mathbf{H}$. If $\alpha >{\lambda }_1(V)$, then for any $\beta \in \mathbb{R}$, ${\displaystyle \underset{u\in \mathbf{H}}{\inf }{J}_{\alpha ,\beta }(u)=-\mathrm{\infty }}$. When $\alpha ={\lambda }_1(V)$, the situation becomes complicated: if $\beta =0$, the corresponding equation is $-\mathrm{\Delta }u={\lambda }_1(V)u$ which has a solution in $\mathbf{H}$ obviously; if $\beta >0$, then ${\displaystyle \underset{u\in \mathbf{H}}{\inf }{J}_{\alpha ,\beta }(u)=-\mathrm{\infty }}$; if $\beta <0$, $J_{\alpha ,\beta }(u)$ has a minimizer in some subspace of $\mathbf{H}$. Moreover, we consider the same problem where higher eigenvalues are involved.