Existence and uniqueness of solutions to Bogomol’nyi-Prased-Sommerfeld equations on graphs

Vortices involve momentous roles in multitudinous fields of theoretical physics comprizing quantum Hall effect, condensed-matter physics, electroweak theory, superconductivity theory, optics, and cosmology. Taubes established the multiple vortex static solutions for the Abelian Higgs model for the first time in [29, 30, 38]. After that, a large number of work related to vortex equations has been accomplished; see, for example, [3, 14, 31, 33, 36] and the references therein. Recently, Lin and Yang pursued a systematic research [21, 22] of the multiple vortex equations obtained in [6, 7, 8, 12, 25, 26, 27, 28]. They established a great deal of sharp existence and uniqueness theorems. The approach they employed include a priori estimates, monotone iterations, and a degree-theory argument, and constrained minimization. In [37], Yang and Lieb presented a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations derived in [5,6], which provide an essential mechanism for linear confinement. Recently, Han [14] established the existence of multipe vortex solutions for the BPS equations derived in [24] from the theory of multi-intersection of D-branes. Chen and Yang [5] proved two sharp existence theorems for the non-Abelian BPS vortex equations arising in the supersymmetric $U(1)\times SU(N)$ gauge theory. In this paper, we investigate the existence and uniqueness of the solutions to the two BPS equations in [5, 14] on a connected finite graph.

During the recent time, the investigation on the equations on graphs has drawn a following and considerable attention from scholars; see, for example, [2, 10, 11, 13, 16, 17, 18, 19, 20, 23, 32, 34, 35] and the references therein. Grigor’yan, Lin and Yang [13] studied the Kazdan-Warner equation

on graph. Huang, Lin and Yau [19] proved the existence of solutions to mean field equations

and

on graphs. Huang, wang and Yang [18] studied the Mean field equation

and the relativistic Ablian Chern-Simons equations

on the finite connected graphs. For more research on Chern-Simons equations on graphs, we refer the readers to [16, 17, 15, 20] and references therein.

The paper is organized as follows. In Section 2, we introduce preliminaries and then state our main results. Section 3 is devoted to the proof of Theorem 2.1. In section 4, we give the proof of Theorem 2.2.

Let $G=(V,E)$ be a connected finite graph, where $V$ denotes the vetex set and $E$ denotes the edge set. Let $\mu:V\to(0,+\infty)$ be a finite measure, and $|V|$=$\text{Vol}(V)=\sum\limits_{x\in V}\mu(x)$ be the volume of $V$. Denote the space of real-valued functions on $V$ by $V^{\mathbb{R}}$. For $x,y\in V$, let $xy$ be the edge from $x$ to $y$. We write $y\sim x$ if $xy\in E$. Let $\omega:V\times V\to[0,\infty)$ be an edge weight function satisfying $\omega_{xy}=\omega_{yx}$ for all $x,y\in V$ and $\omega_{xy}>0$ iff $x\sim y$. For any function $u:V\to\mathbb{R}$, the Laplacian of $u$ is defined by

The gradient $\nabla$ of function $f$ is defined by a vector

The associated gradient form reads

We denote the length of the gradient of $u$ by

Denote, for $p>0$ and any $u\in V^{\mathbb{R}}$, an integral of $u^{p}$ on $V$ by $\int\limits_{V}u^{p}d\mu=\sum\limits_{x\in V}\mu(x)u^{p}(x)$.

In this paper, we study two BPS equations on $G$. One is

$l$, $N_{j}(j=1,\dots,l)$ are positive integers, $p_{j,s}(j=1,\dots,l,s=1,\dots,N_{j})$ are arbitrarily chosen distinct vertices on the graph, and $\delta_{p}$ is the Dirac mass at the vertex $p$. The other is

where

$n_{i}(i=1,\dots,N)$ are positive integers, $\theta,e,g$ are constants and $N$ is a positive integer.

Our main results can be stated as following:

As in [13], we define a sobolev space and a norm by

and

We next give the following Sobolev embedding and Poincaré inequality which will be used later in the paper.

Hereafter, we use boldfaced letters to denote column vectors and $A^{T}$ to denote the transpose of $A$ for any matrix $A$.

Since $\int\limits_{V}4\pi\sum\limits_{s=1}^{N_{j}}\delta_{p_{j,s}}d\mu=4\pi N_{j}$, $j=1,2,\cdots,l$, by a similar arguments as in the proof of Lemma 2.4 of [17], we can show that there exists $u_{j}^{0}$ such that

and that $(u_{1}^{0},u_{2}^{0},\dots,u_{l}^{0})^{T}$ is the unique solution of (7) by up to a constant vector $\mathbf{C}$. Suppoe $\mathbf{u}=(u_{1},u_{2},\cdots,u_{l})$ is a solution to (3). Set $v_{j}:=u_{j}-u_{j}^{0}$, $j=1,2,\cdots,l$. Then we know that

The following lemma gives a necessary condition for $\eqref{11}$ to have a solution.

The following lemma gives a sufficient condition for (3) to have a solution.

Set $u_{i}^{0}$ be a solution of

Set

Then (4) is transformed into

We now write (36) in the vector form

where

and $H=\left(a_{ij}\right)_{N\times N}$ is a $N\times N$ matrix. In view of (5), one may check that $H$ is positive definite. Then, applying the Cholesky decomposition theorem, we can write

where $S=(t_{ij})_{N\times N}$ is an upper triangular matrix,

$a=\frac{\left(e^{2}/2-g^{2}/2N\right)}{N}$ and $b=\frac{g^{2}}{2N}$. Set $\mathbf{v}=\left(v_{1},\ldots,v_{N}\right)^{T},L=\left(S^{T}\right)^{-1}=:\left(l_{ij}\right)_{N\times N}$ and

Combining this with (37), one may obtain

Taking $\alpha_{N}=0$, we rewrite (41) as

Here, we understand $\sum_{j=N+1}^{N}$ as $\sum_{j=N}^{N}$ when $i=N$. Define the energy functional

Then it is easy to check that the system of equations (42) are the Euler Lagrange equations of the functional $J$. The following lemma gives a necessary condition for (4) to admit a solution.