p-Laplacian Equations on Locally Finite Graphs

This paper is mainly concerned with the following nonlinear p-Laplacian equation -Delta(p)u(x) + (lambda a(x) + 1)vertical bar u vertical bar(p-2)(x)u(x) = f(x,u(x)), in V on a locally finite graph G =(V, E) with more general nonlinear term, where Delta(p) is the discrete p-Laplacian on graphs, p >= 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution u(lambda) via the method of Nehari manifold, for any lambda > 1. In addition, as lambda -> + infinity, we prove that the solution u(lambda) converge to a solution of the following Dirichlet problem {-Delta(p)u(x) + vertical bar u vertical bar|(p-2)(x)u(x) = f(x,u(x)), in Omega, u(x) = 0, on partial derivative Omega, where omega = {x is an element of V: a(x) = 0} is the potential well and partial derivative Omega denotes the the boundary of Omega.