Topological degree for Kazdan–Warner equation in the negative case on finite graph

Let \(G=\left( V,E\right) \) $G=\left(V,,,E\right)$ be a connected finite graph. We are concerned about the Kazdan–Warner equation in the negative case on G, say \(\begin{aligned} -\Delta u=h_\lambda e^{2u}-c, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u=h_{\lambda }{e}^{2u}-c,\end{array}$$ where \(\Delta \) $\mathrm{\Delta }$ is the graph Laplacian, \(c<0\) $c<0$ is a real constant, \(h_\lambda =h+\lambda \) $h_{\lambda }=h+\lambda$ , \(h:V\rightarrow \mathbb {R}\) $h:V\to \mathbb{R}$ is a function satisfying \(h\le \max _{V}h=0\) $h\le {max}_{V}h=0$ and \(h\not \equiv 0\) $h\not\equiv 0$ , \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ . In this paper, using the method of topological degree, we prove that there exists a critical value \(\Lambda ^*\in (0,-\min _{V}h)\) ${\mathrm{\Lambda }}^{\ast }\in (0,-{min}_{V}h)$ such that if \(\lambda \in (-\infty ,\Lambda ^*]\) $\lambda \in (-\infty ,{\mathrm{\Lambda }}^{\ast }]$ , then the above equation has solutions; and that if \(\lambda \in (\Lambda ^*,+\infty )\) $\lambda \in ({\mathrm{\Lambda }}^{\ast },+\infty )$ , then it has no solution. Specifically, if \(\lambda \in (-\infty ,0]\) $\lambda \in (-\infty ,0]$ , then it has a unique solution; if \(\lambda \in (0,\Lambda ^*)\) $\lambda \in (0,{\mathrm{\Lambda }}^{\ast })$ , then it has at least two distinct solutions, of which one is a local minimum solution; while if \(\lambda =\Lambda ^*\) $\lambda ={\mathrm{\Lambda }}^{\ast }$ , it has at least one solution. For the proof of these results, we first calculate the topological degree of a map related to the above equation, and then we utilize the relationship between the topological degree and the critical group of the relevant functional. Our method is essentially different from that of Liu and Yang (Calc. Var. 59 (2020), 164), who obtained similar results by using a method of variation.