The ground state solutions of nonlinear Schr\"{o}dinger equations with Hardy weights on lattice graphs

TL;DR

This paper investigates the existence and behavior of ground state solutions for a nonlinear Schrödinger equation with Hardy weights on lattice graphs, revealing how solutions behave as a parameter varies.

Contribution

It introduces a new analysis of Schrödinger equations with Hardy weights on lattice graphs, establishing existence and asymptotic properties of ground states.

Findings

Existence of ground state solutions for small Hardy weight parameter.

Asymptotic behavior of solutions as the Hardy weight approaches zero.

Application of the generalized linking theorem to lattice graph equations.

Abstract

In this paper, we study the nonlinear Schr\"{o}dinger equation $$ -\Delta u+(V(x)- \frac{\rho}{(|x|^2+1)})u=f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with $N\geq 3$, where $V$ is a bounded periodic potential and $0$ lies in a spectral gap of the Schr\"{o}dinger operator $-\Delta+V$. Under some assumptions on the nonlinearity $f$, we prove the existence and asymptotic behavior of ground state solutions with small $\rho\geq 0$ by the generalized linking theorem.