The existence of ground state solutions for nonlinear p-Laplacian equations on lattice graphs

Abstract

In this paper, we study the nonlinear $p$-Laplacian equation $$-{\Delta }_{p}u+V(x)|u{|}^{p-2}u=f(x,u)$$ with positive and periodic potential $V$ on the lattice graph ${\mathbb{Z}}^N$, where ${\Delta }_p$ is the discrete $p$-Laplacian, $p\in (1,\infty )$. The nonlinearity $f$ is also periodic in $x$ and satisfies the growth condition $|f(x,u)|\le a(1+|u{|}^{q-1})$ for some $q>p$. We first prove the equivalence of three function spaces on ${\mathbb{Z}}^N$, which is quite different from the continuous case and allows us to remove the restriction $q>p^{\ast }$ in [SW10], where $p^{\ast }$ is the critical exponent for $W^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$ with $\Omega \subset {\mathbb{R}}^N$ bounded. Then, using the method of Nehari [Neh60, Neh61], we prove the existence of ground state solutions to the above equation.

12 pages