A Class of p-Laplacian Equations on Lattice Graphs

Abstract

In this paper, we study the p-Laplacian equation of the form

$$-\Delta_{p}u+h(x)\vert u \vert^{p-2}u=(R_{\alpha}* \vert u \vert^{q})\vert u \vert^{q-2}u+\vert u \vert ^{2q-2}u$$

on lattice graphs ℤ^N, where N ∈ ℕ*, α ∈ (0, N), \(2 \leq p < {2Nq \over N+\alpha}<+\infty\) and R_α represents the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but behaves as the Riesz potential at infinity. Under suitable assumptions on the potential h(x), we prove the existence of ground state solutions to the equation above by two different methods.