Elliptic problems on weighted locally finite graphs

Let G := (V,E) be a weighted locally finite graph whose finite measure \mu has a positive lower bound. Motivated by a wide interest in the current literature, in this paper we study the existence of classical solutions for a class of elliptic equations involving the \mu-Laplacian operator on the graph G. More precisely, by exploiting direct variational methods, we study problems whose simple prototype has the form { –\Delta u(x) = \lambda f (x, u(x)), for x \in int D; u | \partial D = 0, where D is a bounded domain of V such that int D = 0 and \partial D different from 0, the nonlinear term f : DxR to R satisfies suitable structure conditions and \lambda is a positive real parameter. By applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two solutions for the treated problems. We emphasize the crucial role played by the famous Ambrosetti-Rabinowitz growth condition along the proof of the passion theorem and its consequences. Our results improve the general results obtained by A. Grigor'yan, Y. Lin, and Y. Yang.