2023/08/19 by Li, Yi, Zhang, Qianwei · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.10002
Let G=(V, E) be a connected finite graph, h be a positive function on V and λ1(V) be the first non-zero eigenvalue of -Δ. For any given finite measure μ on V, define functionals J β(u)amp;=amp;(1)/(2)∫V|∇ u|2d μ-βlog∫Vheud μ, J α,β(u)amp;=amp;(1)/(2)∫V(|∇ u|2- αu2) d μ-βlog∫Vheud μ on the functional space \bf H= \ u∈\bf W1,2(V) | ∫Vu dμ=0 \. For any β∈ ℝ, we show that J β(u) has a minimizer u∈\bf H, and then, based on variational principle, the Kazdan-Warner equation Δu=-\fracβheu∫Vheud μ+(β)/(Vol(V)) has a solution in \bf H. If α< λ1(V), then for any β∈ ℝ , J α,β(u) has a minimizer in \bf H, thus the Kazdan-Warner equation Δu+α u=-\fracβheu∫Vheud μ+(β)/(Vol(V)) has a solution in \bf H. If α> λ1(V), then for any β∈ ℝ, inf_u∈\bf H J α,β(u) =- ∞. When α=λ1(V), the situation becomes complicated: if β=0, the corresponding equation is -Δu=λ1(V)u which has a solution in \bf H obviously; if β>0, then inf_u∈ \bf H Jα,β(u) =- ∞; if β<0, J α,β(u) has a minimizer in some subspace of \bf H. Moreover, we consider the same problem where higher eigenvalues are involved.