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

The Chern-Simons theory was originally formulated by Shiing-Shen Chern and James Simons in 1974. This theory was initially developed within the field of mathematics to study the geometric structures on three-dimensional manifolds. Later, it found broad applications in physics, particularly in quantum physics and condensed matter physics. The theory quickly gained significance in understanding topological phase transitions, especially in the context of quantum Hall effects, topological insulators, and high-temperature superconductors [44, 43, 7].

Substantial studies have been undertaken on the dynamic models of the Chern-Simons type in the field theory, referenced in works like [34, 40, 33] among others. From a mathematical perspective, the dynamical equations for different Chern-Simons frameworks pose significant analytical challenges, even in scenarios involving radial symmetry and static conditions [31]. In the Abelian Chern-Simons model, the self-dual structure was discovered in [20, 30] and has catalyzed extensive further investigation. Many problems of existence can be transformed into studies of elliptic partial differential equations or systems of equations, with a particular focus on exploring topological and non-topological solutions [2, 48, 3, 18, 28, 17].

Partial differential equations on discrete graphs have recently garnered significant interest and are now widely applied in diverse fields such as image processing, social network analysis, bioinformatics, and machine learning. Recent advancements have extended some traditional methods for solving partial differential equations in Euclidean spaces to the context of graphs.

Pioneering research by Grigor’yan et al. [15, 14, 16] introduced a variational method to examine the Kazdan-Warner equation, the Yamabe-type equation, and other nonlinear equations. The main goal of these studies was to demonstrate the existence of solutions. For a range of partial differential equations, further investigations have been thoroughly investigated on graphs. Notably, the existence results for Yamabe type equations have been detailed in references such as [12, 10, 51]. Studies addressing Kazdan-Warner equations are found in [50, 9, 11, 46, 36], and the results concerning Schrödinger equations appear in [49, 39, 42, 4]. Additionally, significant findings related to the heat equations are documented in [35, 37].

Define $\Delta$ as the Laplacian operator. In this research, we analyze a generalized Chern-Simons equation described by

where $\lambda$ is a positive constant, $p$ is a non-negative integer, $n_{i}$ for $1\leq i\leq M$ are positive integers, and $p_{i}$ for $1\leq i\leq M$ are specific points with $\delta_{p_{j}}$ denoting the Dirac delta function at $p_{j}$. Within Euclidean spaces, the classification of a solution $u(x)$ to equation (1.1) depends on its behavior at infinity: if $u(x)$ approaches 0 as $|x|$ increases indefinitely, it is termed topological, and if $u(x)$ declines to $-\infty$ as $|x|$ increases, it termed non-topological.

For $p=0$ and $n_{i}=1$ for all $i=1,2,\dots,M$, Eq.(1.1) simplifies to

This equation was examined by Caffarelli et al. [1] and Tarantello [47] within a doubly periodic setting or on the 2-torus in $\mathbb{R}^{2}$, confirming solution presence. Similarly, Huang et al. [26] and Hou et al. [23] explored Eq.(1.2) on finite graphs, establishing solution existence. For $p=2$ and $n_{i}=1$ for each $i=1,2,\dots,M$, Eq.(1.1) modifies to

Han [19] determined the presence of multi-vortices for Eq.(1.3) in a similarly doubly periodic region of $\mathbb{R}^{2}$. Chao et al. [5] and Hu [24] documented multiple solution findings for Eq.(1.3) on finite graphs. Additional research on Chern-Simons models on graphs has been reported in [38, 22, 21, 8, 6, 32, 25, 29].

We describe the lattice $\mathbb{Z}^{n}$ for $n\geq 2$ as a discrete graph denoted by $\mathbb{Z}^{n}=(V,E)$. Here, $V$ denotes the set of vertices and $E$ the set of edges:

where

Consider a finite subset $\Omega\subset V$. The boundary, denoted $\delta\Omega$, is defined as

and we denote by $\bar{\Omega}=\Omega\cup\delta\Omega$ the closure of $\Omega$.

Now, we introduce function spaces and operators on graphs to prepare for the subsequent analysis. Let $C(V)$ denote the collection of all functions on $V$. Similarly, for a finite subset $\Omega$ of $V$, the set $C(\Omega)$ is defined as the collection of functions that are defined over $\Omega$. For any $u\in C(V)$, the integration of $u$ on V is defined as

Analogously, we define the $l^{q}$-norm $(1\leq q<\infty)$ of $u\in C(V)$ by

and the $l^{\infty}$-norm of $u\in C(V)$ by

Denote $x\sim y$ whenever $xy$ is an edge in $E$. We also define the following seminorm:

The Laplacian operator for any $u\in C(V)$ is identified as

Furthermore, for any $u$ and $v$ on the graph, we characterize the gradient form as follows:

If $u=v$, this simplifies to $\Gamma(u)=\Gamma(u,u)$.

For functions $u,v$ in $C(\bar{\Omega})$, a bilinear form $E_{\Omega}(u,v)$ is specified as

and $E_{\Omega}(u)=E_{\Omega}(u,u)$, referred to as the Dirichlet energy of $u$. Lemma 2.2 in [49] indicates that for $u\in C(V)$ and $v\in C(\bar{\Omega})$ with $v$ vanishing on $\partial\Omega$, the following equality holds:

Let $d(x)=d(x,0)$ represent distance from the origin. In this study, we focus primarily on global solutions to the following equation defined on $V$:

Our goal is to establish a topological solution which is also maximal. The primary finding is presented as follows:

In our proof, we employ methods from [45, 25]. First, we establish an iterative scheme, which yields a monotone sequence $\{u_{k}\}$ addressing the Dirichlet problem. Subsequently, we introduce the functional $J_{\Omega}(u)$ and demonstrate that $J_{\Omega}(u_{k})$ is uniformly bounded. By using the discrete Gagliardo-Nirenberg-Sobolev inequality, we ensure that the sequence $\{\|u_{k}\|_{l^{2p+2}(\Omega)}\}$ is uniformly bounded. As we pass to the limit, we ascertain a solution on $\Omega$. Utilizing the exhaustion method, we extend this solution to $V$. Additionally, we establish the maximality of the solution.

Initially, we recall the maximum principle. It a foundational result on graphs.

We then apply the maximum principle to construct an iterative sequence. Define $\Omega_{0}$ as a finite subset of $V$ encompassing $\{p_{1},p_{2},\ldots,p_{M}\}$. Additionally, define $\Omega$ as a connected, finite subset such that $\Omega_{0}\subset\Omega\subset V$. Set $h=4\pi\sum_{j=1}^{M}n_{j}\delta_{p_{j}}$ and define $N=4\pi\sum_{j=1}^{M}n_{j}$. Select a constant $\Lambda>(2p+2)\lambda>0$, initialize $u_{0}=0$, and proceed with the iterative scheme:

Next, we define the natural functional $J_{\Omega}(u)$ and demonstrate that $J_{\Omega}(u_{k})$ decreases as $k$ increases. We introduce the following functional defined over $\Omega$:

Next, by applying the discrete Gagliardo-Nirenberg-Sobolev inequality and invoking Lemma 2.3, we determine the upper bound for $\|u_{k}\|_{l^{2p+2}(\Omega)}$. The foundation for this application is drawn from the proof presented in Theorem 4.1 in [41]:

Lemma 2.1 in [27] implies that

for any $p^{\prime\prime}\geqslant p^{\prime}$. Letting $\gamma=2(p+1)$, $q=2$ and $q^{\prime}=2$, we have for $u\in l^{2}(V)$,