Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs

This work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph (V,E,μ,w): (−Δ)su=(K+λ)e2u−κinV, where the fraction s∈(0,1) and real parameter λ are given, and the graph functions K and κ satisfy maxx∈VK(x)=0, K≢0 and ∫Vκdμ<0. We derive the solvability characteristics of the above equation with the help of variational theory and the upper and lower solutions method.