Poisson problem with Hardy operator on lattice graph ${\mathbb{Z}}^d$ and the application to semilinear equations

In this paper, we investigate the analytic properties of the Hardy-type operator $-\mathrm{\Delta }+\mu H_0$ on the $d$-dimensional integer lattice graph ${\mathbb{Z}}^d$, where $\mu \ge -1$ and $H_0$ is a critical Hardy potential at infinity–arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green kernel estimates on ${\mathbb{Z}}^d$ and obtain a necessary and sufficient weighted summability condition characterizing the solvability of the nonnegative Poisson problem associated with this operator. Building upon this classification, we provide a complete characterization of the nonexistence of positive solutions to the semilinear elliptic inequality $$-\mathrm{\Delta }u+\mu H_{0}u\ge W{u}^p\text{in }{\mathbb{Z}}^d,$$ where $p>0$, $d\ge 3$, $\mu \ge -1$, and $H_0,W\in C({\mathbb{Z}}^d)$ are strictly positive potentials satisfying the asymptotic behaviors $H_0(x)\sim \mathrm{\mid }x{\mathrm{\mid }}^{-2}$ and $W(x)\sim \mathrm{\mid }x{\mathrm{\mid }}^{\theta }$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, with $\theta \in \mathbb{R}$.