The Ground State Solutions of Discrete Nonlinear Schrödinger Equations with Hardy Weights

Abstract

In this paper, we study the discrete nonlinear Schrödinger equation

$$\begin{aligned} -\Delta u+\left( V(x)- \frac{\rho }{(|x|^2+1)}\right) u=f(x,u),\quad u\in \ell ^2({\mathbb {Z}}^N), \end{aligned}$$

where \(N\ge 3\), V is a bounded periodic potential and 0 lies in a spectral gap of the Schrödinger operator \(-\Delta +V\). The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is the lack of compactness of the Cerami sequence. We overcome these two major difficulties by the generalized linking theorem and Lions lemma. This enables us to establish the existence and asymptotic behavior of ground state solutions for small \(\rho \ge 0\).