Existence results for Kazdan–Warner type equations on graphs

In this paper, motivated by the work of Huang et al., Commun. Math. Phys. 377, 613–621 (2020), Sun and Wang, Adv. Math. 404, 108422 (2022), and Li et al., Calc. Var. Partial Differ. Equ. 63, 81 (2024), we investigate the existence of solutions to Kazdan–Warner type equations on a finite connected graph, which are based on the theory of Brouwer degree. Specifically, we consider the equation −Δu = h(x)f(u) − c, where h is a real-valued function defined on the vertex set V, c∈R and f satisfies the asymptotic condition f(t)et→1 as t → ±∞. Different from the above studies, the main difficulty in this paper is to show that the Kazdan–Warner type equation has only three constant solutions, based on delicate analysis and the connectivity of graphs, which have not been extensively explored in previous literature.