Multiple solutions for a generalized Chern-Simons equation on graphs

Abstract

In this paper, we consider a generalized Chern-Simons equationΔu=λeu(eu−1)5+4π∑j=1Nδpj on a connected finite graph G=(V,E), where λ denotes a positive constant; N denotes a positive integer; p1,p2,⋅⋅⋅,pN denote distinct vertices of V; δpj denotes the Dirac delta mass at pj. Using the upper-lower solution method and prior estimates, we prove that there exists a critical value λc such that the generalized Chern-Simons equation admits a solution if λ≥λc. Then applying the mountain pass theorem due to Ambrosetti-Rabinowitz, we establish that the equation has at least two solutions if λ>λc.