Existence of ground state solutions for the nonlinear elliptic equations with zero mass on lattice graphs

Abstract

In this paper, we will study the following nonlinear elliptic equation with zero mass on the lattice graph \begin{equation}\label{A}\tag{A} \begin{cases} -\Delta_{p} u= K(x)f(u) & \hbox{in } \mathbb{Z}^N, \\ u\in D^{1, p}\big(\mathbb{Z}^N\big), \end{cases} \end{equation} where $ N\geq 3$, $1< p< N$, $K$ is a nonnegative potential function, $f$ is a continuous function with quasicritical growth or supercritical growth. By employing variational methods, we establish the existence of ground states for the above equation with an asymptotically periodic potential and vanishing potential at infinity. For the case of asymptotically periodic potential, we also generalize the main result from $\mathbb{Z}^N$ to quasi-transitive graphs.