On topological solutions to a generalized Chern-Simons equation on lattice graphs

TL;DR

This paper proves the existence of a topological solution to a generalized Chern-Simons equation on lattice graphs in higher dimensions, extending previous results and employing the exhaustion method.

Contribution

It introduces a new existence proof for topological solutions to the generalized Chern-Simons equation on z^n, expanding the mathematical understanding of such equations.

Findings

Existence of a global topological solution on z^n.

Extension of previous results to higher dimensions.

Application of the exhaustion method for proof.

Abstract

For $n \geq 2$, consider $\mathbb{Z}^n$ as a lattice graph. We explore a generalized Chern-Simons equation on $\mathbb{Z}^n$. Employing the method of exhaustion, we prove that there exists a global solution that also qualifies as a topological solution. Our results extend those of Hua et al. [arXiv:2310.13905] and complement the findings of Chao and Hou [J. Math. Anal. Appl. $\bf{519}$(1), 126787(2023)], as well as those of Hou and Qiao [J. Math. Phys. $\bf{65}$(8), 081503(2024)].