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 $-Δ+ μH_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $μ\geq -1$ and $H_0$ is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pol