Multiple solutions for ageneralized Chern-Simons equation on graphs

In this paper, we consider a generalized Chern-Simons equation Delta u = lambda e(u) (e(u) - 1)(5) + 4 pi Sigma(N)(j=1)delta(pj) on a connected finite graph G = (V, E), where lambda denotes a positive constant; N denotes a positive integer; p(1), p(2), center dot center dot center dot, p(N) denote distinct vertices of V; delta(pj) denotes the Dirac delta mass at p(j). Using the upper-lower solution method and prior estimates, we prove that there exists a critical value lambda(c) such that the generalized Chern-Simons equation admits a solution if lambda >= lambda(c). Then applying the mountain pass theorem due to Ambrosetti-Rabinowitz, we establish that the equation has at least two solutions if lambda > lambda(c). (c) 2022 Elsevier Inc. All rights reserved.