In this paper, we study the nonlinear p-Laplacian equation \(-\Delta_{p}u+V(x)|u|^{p-2}u=f(x,u)\) on the lattice graph ℤN, where Δp is the discrete p-Laplacian, p ∈ (1, ∈). We study two cases of potential V. The first is periodic and the second tends to a constant at infinity, both of which are positive and bounded. The nonlinearity f is superlinear and satisfies the growth condition ∣f(x, u)∣ ≤ a(1 + ∣u∣q−1) for some q >p. We first prove the equivalence of three function spaces on ℤN, which is quite different from the continuous case and allows us to remove the restriction q > p* in [Handbook of Nonconvex Analysis and Applications, 597–632, Somerville, MA: Int. Press, 2010], where p* is the Sobolev critical exponent. Then, using the method of Nehari ([Trans. Amer. Math. Soc., 1960, 95: 101–123] and [Acta Math., 1961, 105: 141–175]), we prove the existence of ground state solutions to the above equation.
