# Ground state solutions to some Indefinite Nonlinear Schr\"{o}dinger equations on lattice graphs

## TL;DR

This paper establishes the existence of ground state solutions for a class of indefinite nonlinear Schrödinger equations on lattice graphs, using a generalized Nehari manifold approach under periodic and spectral gap conditions.

## Contribution

It introduces a novel application of the generalized Nehari manifold method to indefinite Schrödinger equations on lattice graphs with periodic potentials.

## Key findings

- Existence of ground state solutions under specified conditions.
- Application of the generalized Nehari manifold method to lattice graph equations.
- Solutions are obtained in the presence of spectral gaps and indefinite potentials.

## Abstract

In this paper, we consider the Schr\"odinger type equation $-\Delta u+V(x)u=f(x,u)$ on the lattice graph $\mathbb{Z}^{N}$ with indefinite variational functional, where $-\Delta$ is the discrete Laplacian. Specifically, we assume that $V(x)$ and $f(x,u)$ are periodic in $x$, $f$ satisfies some growth condition and 0 lies in a spectral gap of $(-\Delta + V)$. We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced in arXiv:1801.06872.