Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

Abstract

In this paper, we consider the $(p,q)$-Laplacian Choquard equation on a finite weighted lattice graph $G=(K^N,E,\mu ,\omega )$, namely for any $1<p<q<N$, $r>1$ and $0<\alpha <N$, $-{\Delta }_{p}u-{\Delta }_{q}u+{V\left(x\right)\left(\right|u|}^{p-2}u+{\left|u\right|}^{q-2}u)$$=\left({\displaystyle \sum _{y\in K^N,y\ne x}}{\displaystyle \frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{r-2}u,$ where ${\Delta }_{\nu }$ is the discrete $\nu$-Laplacian on graphs, and $\nu \in \{p.q\}$, $V\left(x\right)$ is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

1. Introduction

Let $G=(V,E)$ be a graph, where V and E, respectively, represent the vertex set and edge set. For two vertices x and y, if there exists an edge e connecting them, i.e., $\{x,y\}\in e$, then we say that they are neighbors, represented by $x\sim y$. The N-dimensional integer lattice graph consists of the set of vertices

$$V={\mathbb{Z}}^N=\{x=(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^N:x_i\in \mathbb{Z},\forall i=1,2,\cdots ,N\}$$

and the set of edges

$$E=\left\{\{x,y\}:x,y\in {\mathbb{Z}}^N,\sum _{i=1}^N|x_i-y_i|=1\right\}.$$

For a fixed integer $K>0$, let $(K^N,E{|}_{K^N})$ be a finite subgraph of $({\mathbb{Z}}^N,E)$, where

$$K^N=\{x\in {\mathbb{Z}}^N:0\le x_i\le K,\forall i=1,2,\cdots ,N\}.$$

Clearly, it is a bounded set and contains $K^N$ vertices. $\forall x,y\in K^N$, and there exists a shortest path $\gamma$ connecting x and y. Let $d(x,y)$ be the distance between x and y, which represents the number of edges belonging to the shortest path $\gamma$. Obviously, if x and y are neighbors, then $d(x,y)=1$. Moreover, $\forall x,y\in K^N$, and it always holds that $d(x,y)\ge 1$. Let $\mu :K^N\to {\mathbb{R}}^{+}$ be a positive finite measure and ${\omega :E|}_{K^N}\to \mathbb{R}$ be a positive symmetric weight. The quadruple $G=(K^N,E,\mu ,\omega )$ is described as a finite weighted lattice graph.

For each function $u:V\to \mathbb{R}$, we define the $\mu$-Laplacian of u as follows:

$$\Delta u\left(x\right):={\displaystyle \frac{1}{\mu \left(x\right)}}\sum _{y\sim x}{\omega }_{xy}(u\left(y\right)-u\left(x\right)).$$

The relevant gradient form is defined by

$$\mathsf{\Gamma }(u,v)\left(x\right):=\nabla u\left(x\right)\nabla v\left(x\right)={\displaystyle \frac{1}{2\mu \left(x\right)}}\sum _{y\sim x}{\omega }_{xy}(u\left(y\right)-u\left(x\right))(v\left(y\right)-v\left(x\right)).$$

The length of the gradient of u is represented by

$$|\nabla u|\left(x\right):=\sqrt{\mathsf{\Gamma }(u,u)\left(x\right)}={\left({\displaystyle \frac{1}{2\mu \left(x\right)}}\sum _{y\sim x}{\omega }_{xy}{(u\left(y\right)-u\left(x\right))}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

(1)

We use $C\left(K^N\right)$ to represent the space composed of all real-valued functions on $K^N$. Regarding vertex weight $\mu$, the integral of $u\in C\left(K^N\right)$ is defined as

$${\int }_{K^N}ud\mu =\sum _{x\in K^N}\mu \left(x\right)u\left(x\right).$$

Through variational calculations on local finite graphs, Grigor’yan et al. [1] have defined the p-Laplacian of $u:V\to \mathbb{R}$ as

$${\Delta }_{p}u\left(x\right):={\displaystyle \frac{1}{2\mu \left(x\right)}}\sum _{y\sim x}{\left(\right|\nabla u|}^{p-2}\left(y\right)+{|\nabla u|}^{p-2}\left(x\right)){\omega }_{xy}(u\left(y\right)-u\left(x\right)).$$

(2)

The study of differential equations on graphs represents a significant mathematical topic, possessing extensive applications across various domains, including data analysis, neural networks and image processing [2,3,4,5]. Recently, many researchers have focused on and studied differential equations on graphs, such as the Kazdan Warner equations, the Yamabe-type equations, the Schr$\ddot{\mathrm{o}}$dinger equations and the Choquard equations; see, for example, [6,7,8,9,10,11,12,13,14,15] and the references therein.

On lattice graphs, Wang et al. in [16] considered the following Choquard equation:

$$-\Delta u+V\left(x\right)u=\left(\sum _{y\in {\mathbb{Z}}^N,y\ne x}{\displaystyle \frac{{\left|u\left(y\right)\right|}^p}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{p-2}u,\mathrm{in}{\mathbb{Z}}^N.$$

(3)

The ground state solution was obtained using the Nehari method. In [17], Han and Shao investigated the nonlinear p-Laplacian equations on locally finite graphs

$$-{\Delta }_{p}u+(\lambda a\left(x\right)+1){\left|u\right|}^{p-2}u=f(x,u)\mathrm{in}V,$$

(4)

where $p\ge 2$. By using the mountain pass theorem and via the method of the Nehari manifold, they proved the existence of ground state solutions. Very recently, Liu and Zhang in [18] investigated the existence of ground state solutions for the p-Laplacian Choquard equation on a finite lattice graph as follows:

$$-{\Delta }_{p}u+{V\left(x\right)\left|u\right|}^{p-2}u=\left(\sum _{y\in N^n,y\ne x}{\displaystyle \frac{{\left|u\left(y\right)\right|}^q}{d{(x,y)}^{n-\alpha }}}\right){\left|u\right|}^{q-2}u.$$

(5)

In a Euclidean space, Xie et al. in [19] studied a class of $(p,q)$-Laplacian equations with a nonlocal Choquard equation. By applying the Nehari method and the Pokhozhaev identity, Guo and Zhao in [20] proved the existence of a ground state solution for the fractional Choquard equations with doubly critical exponents and magnetic fields. In [21], Chen and Wang investigated a class of biharmonic Choquard equations with exponential critical growth in ${\mathbb{R}}^4$.

However, as far as we know, there are no results for Choquard equations with $(p,q)$-Laplacians on graphs. Motivated by the aforementioned works, we investigate the existence of mountain pass solutions and ground state solutions for the nonlinear $(p,q)$-Laplacian Choquard equation on a finite weighted lattice graphs G as follows:

$$-{\Delta }_{p}u-{\Delta }_{q}u+{V\left(x\right)\left(\right|u|}^{p-2}u+{\left|u\right|}^{q-2}u)=\left(\sum _{y\in K^N,y\ne x}{\displaystyle \frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{r-2}u,$$

(6)

where ${\Delta }_{k}u$, with $k\in \{p.q\}$, is the k-Laplacian operator; the potential V is a positive function on G; and $d(x,y)$ represents the distance between x and y. The feature of Equation (6) includes the combined effects of a Choquard reaction and a double-phase operator on graphs.

In this paper, we investigate the existence of solutions for Equation (6). The main results obtained are based on variational methods and some analytical techniques. The structure of this article is as follows. In Section 2, we introduce some notations and preliminaries. Section 3 is dedicated to proving our main results. Lastly, Section 4 summarizes the conclusions of this paper.

2. Preliminaries

For any $1\le q\le +\infty$, the $l^q\left(K^N\right)$ space on a finite weighted lattice graph $K^N$ is

$$l^q\left(K^N\right):=\left\{u\in C\left(K^N\right):{\left|u\right|}_q<+\infty \right\},$$

where the norm of $u\in l^q\left(K^N\right)$ is given as

$${\left|u\right|}_q={\left(\underset{K^N}{\int }{\left|u\right|}^{q}d\mu \right)}^{{\displaystyle \frac{1}{q}}}=\left\{\begin{array}{cc}{\left({\displaystyle \sum _{x\in K^N}}\mu \left(x\right){\left|u\left(x\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}},\hfill & 1\le q<+\infty ,\hfill \\ \underset{x\in K^N}{sup}\mu \left(x\right)\left|u\left(x\right)\right|,\hfill & q=+\infty .\hfill \end{array}\right.$$

Define

$$W^{1,p}\left(K^N\right):=\{u\in C\left(K^N\right){:\parallel u\parallel }_{W_{1,p}\left(K^N\right)}<+\infty \}.$$

Here,

$${\parallel u\parallel }_{W_{1,p}\left(K^N\right)}={\left(\underset{K^N}{\int }{|\nabla u|}^p+{\left|u\right|}^{p}d\mu \right)}^{{\displaystyle \frac{1}{p}}}.$$

Let

$$X:=\left\{u\in W^{1,p}\left(K^N\right)\cap W^{1,q}\left(K^N\right):\underset{K^N}{\int }V\left(x\right){\left(\right|u|}^p+{\left|u\right|}^q)d\mu <+\infty \right\}$$

equipped with the norm $\parallel u\parallel ={\parallel u\parallel }_{p,V}+{\parallel u\parallel }_{q,V}$, where

$${\parallel u\parallel }_{\nu ,V}={\left(\underset{K^N}{\int }{|\nabla u|}^{\nu }+V\left(x\right){\left|u\right|}^{\nu }d\mu \right)}^{{\displaystyle \frac{1}{\nu }}},\nu \in \{p,q\}.$$

The energy functional $J_{p,q}:X\to \mathbb{R}$ associated with Equation (6) is provided by

$$J_{p,q}\left(u\right)={\displaystyle \frac{1}{p}}{\parallel u\parallel }_{p,V}^p+{\displaystyle \frac{1}{q}}{\parallel u\parallel }_{q,V}^q-{\displaystyle \frac{1}{2r}}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}{\displaystyle \frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{r}d\mu .$$

(7)

It is easy to see that $J_{p,q}\in C^1(X,\mathbb{R})$ has the Fréchet derivative given by

$$\begin{array}{cc}{\displaystyle J_{p,q}^{\prime }\left(u\right)\left(\varphi \right)}\hfill & =\underset{K^N}{\int }{\left[\right|\nabla u|}^{p-2}\mathsf{\Gamma }(u,\varphi )+{|\nabla u|}^{q-2}\mathsf{\Gamma }(u,\varphi )]d\mu \hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }{V\left(x\right)\left[\right|u|}^{p-2}u+{\left|u\right|}^{q-2}u]\varphi d\mu -\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r-2}u\varphi d\mu ,\forall \varphi \in X.}\hfill \end{array}$$

(8)

Definition 1 

([22]). Let $(E,\parallel ·\parallel )$ be a Banach space, $J\in C^1(E,\mathbb{R})$. We say that the function J satisfies the Palais Smale sequence (${\left(PS\right)}_c$) condition if any $\left\{u_n\right\}\subset E$ such that $J\left(u_n\right)\to c$ and $J^{\prime }\left(u_n\right)\to 0$ as $n\to +\infty$ has a convergent subsequence.

Based on the fact that $V\left(x\right)\ge V_0={min}_{x\in K^N}V\left(x\right)>0$, similar to the proof of Lemma 3.4 in [23], we can obtain the following embedding property.

Lemma 1. 

X embeds continuously into $L^{\nu }\left(K^N\right)$ for $\nu \in [p,q^{*}]$ and compactly into $L^{\nu }\left(K^N\right)$ for $\nu \in [p,q^{*})$. Moreover, there exists a constant ${\chi }_{\nu }>0$ such that

$${\displaystyle {\left|u\right|}_{\nu }\le {\chi }_{\nu }\parallel u\parallel ,\mathrm{for}\mathrm{all}\nu \in [p,q^{*}],}$$

(9)

where $q^{*}=qN/(N-q)$.

To deal with the Choquard equation, we need the following discrete Hardy–Littlewood–Sobolev inequality.

Lemma 2 

([24]). Let $G=({\mathbb{Z}}^N,E,\mu ,\omega )$ be a weighted lattice graph satisfying the following condition:

$\left(G1\right)$ the measure μ is uniformly bounded, i.e., there are positive constants ${\mu }_0$ and ${\mu }_1$ that satisfy ${\mu }_0\le \mu \left(x\right)\le {\mu }_1$ for all $x\in {\mathbb{Z}}^N$.

Assume that $0<\alpha <N$, $1<h,k<+\infty$ and $1/h+1/k+(N-\alpha )/N=2$. If $u\in l^h\left(K^N\right)$, $v\in l^k\left(K^N\right)$, then there exists a positive constant C depending only on $N,h,k$ and α such that

$$\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}{\displaystyle \frac{v\left(y\right)}{d{(x,y)}^{N-\alpha }}}\right)u\left(x\right)d\mu \le {C\left|u\right|}_h{\left|v\right|}_k.$$

Remark 1. 

Obviously, a finite weighted lattice graph $(K^N,E,\mu ,\omega )$ satisfies the condition $\left(G1\right)$.

Based on Lemma 2.1 in [17], we can derive the following lemma.

Lemma 3. 

Assume that $u\in X$ and its p-Laplacian is defined by (2). Then, for any $\varphi \in C\left(K^N\right)$, one has

$$\underset{K^N}{\int }{|\nabla u|}^{p-2}\nabla u\nabla \varphi d\mu =\underset{K^N}{\int }{|\nabla u|}^{p-2}\mathsf{\Gamma }(u,\varphi )d\mu =-\underset{K^N}{\int }\left({\Delta }_{p}u\right)\varphi d\mu .$$

If, for any $\varphi \in X$, there holds

$$\begin{array}{cc}{\displaystyle J_{p,q}^{\prime }\left(u\right)\left(\varphi \right)=}\hfill & \underset{K^N}{\int }{\left[\right|\nabla u|}^{p-2}\mathsf{\Gamma }(u,\varphi )+{|\nabla u|}^{q-2}\mathsf{\Gamma }(u,\varphi )]d\mu +\underset{K^N}{\int }{V\left(x\right)\left[\right|u|}^{p-2}u+{\left|u\right|}^{q-2}u]\varphi d\mu \hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r-2}u\varphi d\mu =0,}\hfill \end{array}$$

(10)

then we call the function $u\in X$ a weak solution of Equation (6). For any fixed vertex $x_0\in K^N$, substituting a test function $\varphi :K^N\to \mathbb{R}$ into (10) with

$$\varphi \left(x\right)=\left\{\begin{array}{cc}1,\hfill & x=x_0,\hfill \\ 0,\hfill & x\ne x_0,\hfill \end{array}\right.$$

and $\varphi \in X$, by (10) and Lemma 3, one has

$$\begin{gathered} {\displaystyle -{\Delta }_{p}u\left(x_0\right)-{\Delta }_{q}u\left(x_0\right)+V\left(x_0\right)\left(\right|u\left(x_0\right){|}^{p-2}u\left(x_0\right)+|u\left(x_0\right){|}^{q-2}u\left(x_0\right)) \\ =\left(\sum _{y\in K^N,y\ne x_0}\frac{{\left|u\left(y\right)\right|}^r}{d{(x_0,y)}^{N-\alpha }}\right){\left|u\left(x_0\right)\right|}^{r-2}u\left(x_0\right),} \end{gathered}$$

and thus u is the point-wise solution of (6). Therefore, to solve Equation (6), we only need to find the critical point of $J_{p,q}$.

3. Main Results

Throughout this paper, C or $C_i$ ($i=1,2,\cdots$) denotes some positive constants that may change row by row. We divide this section into two subsections to discuss the existence of the mountain pass solution and the ground state solution separately.

3.1. Existence of Mountain Pass Solution

In the following, we will prove that $J_{p,q}$ satisfies the geometric structure of the mountain pass theorem.

Lemma 4. 

Let $G=(K^N,E,\mu ,\omega )$ be a finite weighted lattice graph. If $1<p<q<N$, and $max\left\{1,{\displaystyle \frac{q}{2}},{\displaystyle \frac{(N+\alpha )p}{2N}}\right\}<r\le {\displaystyle \frac{(N+\alpha )q}{2(N-q)}}$, then

(i) 

there exist $\sigma ,\rho >0$ such that $J_{p,q}\left(u\right)\ge \sigma$, $\forall \parallel u\parallel =\rho$;

(ii) 

there exist $R>0$ and $e\in X$ with $\parallel e\parallel >R$ such that $J_{p,q}\left(e\right)<0$.

Proof. 

(i) Taking $\rho \in (0,1)$, let $u\in X$ with $\parallel u\parallel =\rho$, and one has ${\parallel u\parallel }_{p,V}^q\le {\parallel u\parallel }_{p,V}^p<1$. From this condition, we have

$$\begin{array}{l}max\left\{1,{\displaystyle \frac{q}{2}},{\displaystyle \frac{(N+\alpha )p}{2N}}\right\}<r\le {\displaystyle \frac{(N+\alpha )q}{2(N-q)}}\\ \Rightarrow {\displaystyle \frac{(N+\alpha )p}{2N}}<r\le {\displaystyle \frac{(N+\alpha )q^{*}}{2N}}\Rightarrow p<{\displaystyle \frac{2rN}{N+\alpha }}\le q^{*}\hfill \end{array}$$

(11)

Taking $h=k=2rN/(N+\alpha )$ in Lemma 2, it follows from Lemma 1 and (7) and (11) that

$$\begin{array}{cc}{\displaystyle J_{p,q}\left(u\right)}\hfill & ={\displaystyle \frac{1}{p}}{\parallel u\parallel }_{p,V}^p+{\displaystyle \frac{1}{q}}{\parallel u\parallel }_{q,V}^q-{\displaystyle \frac{1}{2r}}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}{\displaystyle \frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{r}d\mu \hfill \\ \hfill & {\displaystyle \ge \frac{1}{q}{(\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^q{)-C|u|}_{\frac{2rN}{N+\alpha }}^{2r}}\hfill \\ \hfill & {\displaystyle \ge \frac{1}{q}{(\parallel u\parallel }_{p,V}^q+{\parallel u\parallel }_{q,V}^q)-C_1{\parallel u\parallel }^{2r}.}\hfill \end{array}$$

(12)

By using the following elementary inequality

$${\displaystyle a^s+b^s\ge c_s{(a+b)}^s,a,b\ge 0,s>1,(\mathrm{constant}{c}_s\in (0,1)),}$$

(13)

we obtain

$${\displaystyle J_{p,q}\left(u\right)\ge \frac{c_q}{q}{(\parallel u\parallel }_{p,V}+{\parallel u\parallel }_{q,V}{)}^q-C_1{\parallel u\parallel }^{2r}=\frac{c_q}{q}{\parallel u\parallel }^q-C_1{\parallel u\parallel }^{2r}.}$$

(14)

In view of $2r>q$, there exists $\rho >0$ sufficiently small and $\sigma >0$ such that $J_{p,q}\left(u\right)\ge \sigma$, $\forall \parallel u\parallel =\rho$.

(ii) Fix $u_0\in X\backslash \left\{0\right\}$ with $u_0>0$, since $2r>q>p$, and we can deduce that

$$\begin{array}{cc}{\displaystyle J_{p,q}\left(s{u}_0\right)}\hfill & ={\displaystyle \frac{s^p}{p}}\parallel u_0{\parallel }_{p,V}^p+{\displaystyle \frac{s^q}{q}}\parallel u_0{\parallel }_{q,V}^q-{\displaystyle \frac{1}{2r}}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}{\displaystyle \frac{|s{u}_0{|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|s{u}_0\right|}^{r}d\mu \hfill \\ \hfill & {\displaystyle \le C_2(s^p+s^q)-\frac{s^{2r}}{2r}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_0{|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_0\right|}^{r}d\mu \to -\infty }\hfill \end{array}$$

(15)

as $s\to \infty$. Taking $e=s{u}_0$ with s large enough, from (15), we can see that conclusion (ii) holds. This completes the proof. □

Lemma 5. 

Under the conditions of Lemma 4, any (PS) sequence $\left\{u_n\right\}$ of $J_{p,q}$ at level $c>0$ is bounded in X.

Proof. 

Let $\left\{u_n\right\}$ be a (PS) sequence at level $c>0$ for $J_{p,q}$, i.e.,

$${\displaystyle J_{p,q}\left(u_n\right)\to c,\mathrm{and}{J}_{p,q}^{\prime }\left(u_n\right)\varphi \to 0,\forall \varphi \in X,}$$

(16)

where $c:=\underset{\psi \in \mathsf{\Gamma }}{inf}\underset{t\in [0,1]}{max}\mathsf{\Gamma }\left(\psi \left(t\right)\right)$ and $\mathsf{\Gamma }:=\{\psi \in C([0,1],X):\psi \left(0\right)=0,J_{p,q}\left(\psi \left(1\right)\right)<0\}$. From (16) and (8), we obtain

$${\displaystyle \frac{1}{p}\parallel u_n{\parallel }_{p,V}^p+\frac{1}{q}\parallel u_n{\parallel }_{q,V}^q-\frac{1}{2r}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r}d\mu =c+o_n\left(1\right)}$$

(17)

$$\begin{array}{c}\hfill {\displaystyle \left|\underset{K^N}{\int }\left[\right|\nabla u_n{|}^{p-2}\mathsf{\Gamma }(u_n,\varphi )+|\nabla u_n{|}^{q-2}\mathsf{\Gamma }(u_n,\varphi )]d\mu +\underset{K^N}{\int }V\left(x\right)\left[\right|u_n{|}^{p-2}{u}_n+|u_n{|}^{q-2}{u}_n]\varphi d\mu \right.}\\ \hfill {\displaystyle \left.=\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r-2}{u}_n\varphi d\mu \right|=o_n\left(1\right)\parallel \varphi \parallel ,\forall \varphi \in X.}\end{array}$$

(18)

Here, $o_n\left(1\right)\to 0$ as $n\to +\infty$. Taking $\varphi =u_n$ in (18), from (1), we obtain

$${\displaystyle \parallel u_n{\parallel }_{p,V}^p+\parallel u_n{\parallel }_{q,V}^q=\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right)|u_n{|}^{r}d\mu +o_n\left(1\right)\parallel u_n\parallel .}$$

(19)

Combining (17) and (19), we obtain

$${\displaystyle \left(\frac{1}{p}-\frac{1}{2r}\right)\parallel u_n{\parallel }_{p,V}^p+\left(\frac{1}{q}-\frac{1}{2r}\right)\parallel u_n{\parallel }_{q,V}^q=c-o_n\left(1\right)\parallel u_n\parallel +o_n\left(1\right).}$$

(20)

In view of (20), (13) and $q>p>1$, we have

$$\begin{array}{c}{\displaystyle c+1-o_n\left(1\right)\parallel u_n\parallel +o_n\left(1\right)\ge \left(\frac{1}{q}-\frac{1}{2r}\right)(\parallel u_n{\parallel }_{p,V}^p+\parallel u_n{\parallel }_{q,V}^q+1)}\hfill \\ {\displaystyle \ge \left(\frac{1}{q}-\frac{1}{2r}\right)(\parallel u_n{\parallel }_{p,V}^p+c_q(\parallel u_n{\parallel }_{q,V}{+1)}^q)\ge \left(\frac{1}{q}-\frac{1}{2r}\right)(\parallel u_n{\parallel }_{p,V}^p+c_q(\parallel u_n{\parallel }_{q,V}+1{)}^p)}\hfill \\ {\displaystyle \ge \left(\frac{1}{q}-\frac{1}{2r}\right)c_q(\parallel u_n{\parallel }_{p,V}^p+(\parallel u_n{\parallel }_{q,V}{+1)}^p)\ge \left(\frac{1}{q}-\frac{1}{2r}\right)c_q{c}_p(\parallel u_n{\parallel }_{p,V}+\parallel u_n{{\parallel }_{q,V}+1)}^p}\hfill \\ {\displaystyle =\left(\frac{1}{q}-\frac{1}{2r}\right)c_q{c}_p(\parallel u_n{\parallel +1)}^p>\left(\frac{1}{q}-\frac{1}{2r}\right)c_q{c}_p{\parallel u_n\parallel }^p.}\hfill \end{array}$$

Thus, by $p>1$ and $2r>q$, we conclude that $\parallel u_n\parallel$ is bounded in X. □

Lemma 6. 

Under the conditions of Lemma 4, any (PS) sequence $\left\{u_n\right\}$ of $J_{p,q}$ has a convergence subsequence.

Proof. 

By Lemma 5, $\left\{u_n\right\}$ of $J_{p,q}$ is bounded in X. Up to a subsequence, there exists some $u\in X$ such that $u_n\rightharpoonup u$ as $n\to +\infty$ and $u_n\to u$ in $l^{\nu }\left(K^N\right)$ for all $p\le \nu \le q^{*}$ (by Lemma 1). From (16), we have

$$\begin{array}{cc}{\displaystyle o_n\left(1\right)}\hfill & =J_{p,q}^{\prime }\left(u_n\right)(u_n-u)=\underset{K^N}{\int }\left[\right|\nabla u_n{|}^{p-2}\mathsf{\Gamma }(u_n,u_n-u)+V\left(x\right)\left[\right|u_n{|}^{p-2}{u}_n(u_n-u)]d\mu \hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }\left[\right|\nabla u_n{|}^{q-2}\mathsf{\Gamma }(u_n,u_n-u)+V\left(x\right)\left[\right|u_n{|}^{q-2}{u}_n(u_n-u)]d\mu }\hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in {\mathbb{Z}}^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r-2}{u}_n(u_n-u)d\mu }\hfill \\ \hfill & {\displaystyle =\underset{K^N}{\int }\left[\right|\nabla u_n{|}^{p-2}\mathsf{\Gamma }(u_n,u_n)-|\nabla u_n{|}^{p-2}\mathsf{\Gamma }(u_n,u)]d\mu +\underset{K^N}{\int }V\left(x\right)|u_n{|}^{p}d\mu -\underset{K^N}{\int }V\left(x\right){\left|u_n\right|}^{p-2}{u}_{n}ud\mu }\hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }\left[\right|\nabla u_n{|}^{q-2}\mathsf{\Gamma }(u_n,u_n)-|\nabla u_n{|}^{q-2}\mathsf{\Gamma }(u_n,u)]d\mu +\underset{K^N}{\int }V\left(x\right)|u_n{|}^{q}d\mu -\underset{K^N}{\int }V\left(x\right){\left|u_n\right|}^{q-2}{u}_{n}ud\mu }\hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in {\mathbb{Z}}^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r-2}{u}_n(u_n-u)d\mu }\hfill \\ \hfill & {\displaystyle =\underset{K^N}{\int }\left[\right|\nabla u_n{|}^p-|\nabla u_n{|}^{p-2}\nabla u_n\nabla u+V\left(x\right)|u_n{|}^p-V\left(x\right)|u_n{|}^{p-2}{u}_{n}u]d\mu }\hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }\left[\right|\nabla u_n{|}^q-|\nabla u_n{|}^{q-2}\nabla u_n\nabla u+V\left(x\right)|u_n{|}^q-V\left(x\right)|u_n{|}^{q-2}{u}_{n}u]d\mu }\hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_n{\left(y\right)|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r-2}{u}_n(u_n-u)d\mu .}\hfill \end{array}$$

(21)

Similarly, we can obtain that

$$\begin{array}{cc}{\displaystyle o_n\left(1\right)}\hfill & =J_{p,q}^{\prime }\left(u\right)(u_n-u)=\underset{K^N}{\int }{\left[\right|\nabla u|}^{p-2}\nabla u\nabla u_n-{|\nabla u|}^p+V\left(x\right){\left|u\right|}^{p-2}u{u}_n-V\left(x\right){\left|u\right|}^p]d\mu \hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }{\left[\right|\nabla u|}^{q-2}\nabla u\nabla u_n-{|\nabla u|}^q+{V\left(x\right)\left|u\right|}^{q-2}u{u}_n-{V\left(x\right)\left|u\right|}^q]d\mu }\hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r-2}u(u_n-u)d\mu .}\hfill \end{array}$$

(22)

From (21) and (22), we obtain

$$\begin{array}{cc}{\displaystyle o_n\left(1\right)}\hfill & =(J_{p,q}^{\prime }\left(u_n\right)-J_{p,q}^{\prime }\left(u\right))(u_n-u)\hfill \\ \hfill & {\displaystyle =\underset{K^N}{\int }\left[\left(\right|\nabla u_n{|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u)(\nabla u_n-\nabla u)+V\left(x\right)\left(\right|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u)(u_n-u)\right]d\mu }\hfill \\ \hfill & {\displaystyle +\underset{K^N}{\int }\left[\left(\right|\nabla u_n{|}^{q-2}\nabla u_n-{|\nabla u|}^{q-2}\nabla u)(\nabla u_n-\nabla u)+V\left(x\right)\left(\right|u_n{|}^{q-2}{u}_n-{\left|u\right|}^{q-2}u)(u_n-u)\right]d\mu }\hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{|u_n\left(y\right){|}^r)}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r-2}{u}_n(u_n-u)d\mu }\hfill \\ \hfill & {\displaystyle -\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r-2}u(u_n-u)d\mu .}\hfill \end{array}$$

(23)

By the H$\ddot{\mathrm{o}}$lder inequality and Lemmas 1, 2 and 5, we can obtain that

$$\begin{array}{c}\left|\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}{\displaystyle \frac{{\left|u_n\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u_n\right|}^{r-2}{u}_n\left(u_n-u\right)d\mu \right|\hfill \\ \le C{\left(\underset{K^N}{\int }{\left|u_n\right|}^{{\displaystyle \frac{2rN}{N+\alpha }}}d\mu \right)}^{{\displaystyle \frac{N+\alpha }{2N}}}{\left(\underset{K^N}{\int }{\left|u_n\right|}^{{\displaystyle \frac{2(r-1)N}{N+\alpha }}}{\left|u_n-u\right|}^{{\displaystyle \frac{2N}{N+\alpha }}}d\mu \right)}^{{\displaystyle \frac{N+\alpha }{2N}}}\hfill \\ \le C{\left|u_n\right|}_{{\displaystyle \frac{2rN}{N+\alpha }}}^r{\left(\underset{K^N}{\int }{\left|u_n\right|}^{{\displaystyle \frac{2rN}{N+\alpha }}}d\mu \right)}^{{\displaystyle \frac{(N+\alpha )(r-1)}{2rN}}}{\left(\underset{K^N}{\int }{\left|u_n-u\right|}^{{\displaystyle \frac{2rN}{N+\alpha }}}d\mu \right)}^{{\displaystyle \frac{N+\alpha }{2rN}}}\hfill \\ =C{\left|u_n\right|}_{{\displaystyle \frac{2rN}{N+\alpha }}}^{2r-1}{\left|u_n-u\right|}_{{\displaystyle \frac{2rN}{N+\alpha }}}\le C_3\parallel {\left.u_n\right|}^{2r-1}{\left|u_n-u\right|}_{{\displaystyle \frac{2rN}{N+\alpha }}}\hfill \\ =o_n\left(1\right)\hfill \end{array}$$

(24)

Similarly, we can find that

$${\displaystyle \underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r-2}u(u_n-u)d\mu =o_n\left(1\right).}$$

(25)

From (23)–(25), it is easy to see that

$${\displaystyle \underset{K^N}{\int }\left[\left(\right|\nabla u_n{|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u)(\nabla u_n-\nabla u)+V\left(x\right)\left(\right|u_n{|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u)(u_n-u)\right]d\mu =o_n\left(1\right),}$$

(26)

and

$${\displaystyle \underset{K^N}{\int }\left[\left(\right|\nabla u_n{|}^{q-2}\nabla u_n-{|\nabla u|}^{q-2}\nabla u)(\nabla u_n-\nabla u)+V\left(x\right)\left(\right|u_n{|}^{q-2}{u}_n-{\left|u\right|}^{q-2}u)(u_n-u)\right]d\mu =o_n\left(1\right).}$$

(27)

By applying the inequality below

$${\displaystyle {\left(\right|\xi |}^{s-2}\xi -{\left|\eta \right|}^{s-2}\eta )(\xi -\eta )\ge \left\{\begin{array}{cc}{c|\xi -\eta |}^s,\hfill & \mathrm{if}s\ge 2,\hfill \\ c\left(\right|\xi |+{\left|\eta \right|)}^{s-2}{|\xi -\eta |}^2,\hfill & \mathrm{if}1<s<2,\hfill \end{array}\right.\forall \xi ,\eta \in {\mathbb{R}}^N,}$$

(28)

and (26), we have, for the case of $p\ge 2$, that

$$\begin{array}{cc}\hfill & {∥u_n-u∥}_{p,V}^p=\underset{K^N}{\int }\left[{\left|\nabla \left(u_n-u\right)\right|}^p+V\left(x\right){\left|u_n-u\right|}^p\right]d\mu \hfill \\ \hfill & \le C\underset{K^N}{\int }\left[\left({\left|\nabla u_n\right|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u\right)\left(\nabla u_n-\nabla u\right)+V\left(x\right)\left({\left|u_n\right|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)\left(u_n-u\right)\right]d\mu \hfill \\ \hfill & =o_n\left(1\right).\hfill \end{array}$$

(29)

On the other hand, for the case of $1<p<2$, utilizing the boundedness of $\left\{u_n\right\}$, (26) and the H$\ddot{\mathrm{o}}$lder inequality, one can derive that

$$\begin{array}{cc}\hfill & \underset{K^N}{\int }{\left|\nabla \left(u_n-u\right)\right|}^{p}d\mu =\underset{K^N}{\int }{\left|\nabla \left(u_n-u\right)\right|}^p{\left(\left|\nabla u_n\right|+|\nabla u|\right)}^{{\displaystyle \frac{p(p-2)}{2}}}·{\left(\left|\nabla u_n\right|+|\nabla u|\right)}^{{\displaystyle \frac{p(2-p)}{2}}}d\mu \hfill \\ \hfill & \le {\left[\underset{K^N}{\int }{\left|\nabla \left(u_n-u\right)\right|}^2{\left(\left|\nabla u_n\right|+|\nabla u|\right)}^{p-2}d\mu \right]}^{{\displaystyle \frac{p}{2}}}{\left[\underset{K^N}{\int }{\left(\left|\nabla u_n\right|+|\nabla u|\right)}^{p}du\right]}^{{\displaystyle \frac{2-p}{2}}}\hfill \\ \hfill & \le C{\left[\underset{K^N}{\int }\left({\left|\nabla u_n\right|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u\right)\left(\nabla u_n-\nabla u\right)d\mu \right]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & =o_n\left(1\right),\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill & \underset{K^N}{\int }V\left(x\right){\left|u_n-u\right|}^{p}d\mu =\underset{K^N}{\int }V{\left(x\right)}^{{\displaystyle \frac{p}{2}}}{\left|u_n-u\right|}^p{\left(\left|u_n\right|+\left|u\right|\right)}^{{\displaystyle \frac{p(p-2)}{2}}}·V{\left(x\right)}^{{\displaystyle \frac{2-p}{2}}}{\left(\left|u_n\right|+\left|u\right|\right)}^{{\displaystyle \frac{p(2-p)}{2}}}d\mu \hfill \\ \hfill & \le {\left[\underset{K^N}{\int }V\left(x\right){\left|u_n-u\right|}^2{\left(\left|u_n\right|+\left|u\right|\right)}^{p-2}d\mu \right]}^{{\displaystyle \frac{p}{2}}}{\left[\underset{K^N}{\int }V\left(x\right){\left(\left|u_n\right|+\left|u\right|\right)}^{p}d\mu \right]}^{{\displaystyle \frac{2-p}{2}}}\hfill \\ \hfill & \le C{\left[\underset{K^N}{\int }V\left(x\right)\left({\left|u_n\right|}^{p-2}{u}_n-{\left|u\right|}^{p-2}u\right)\left(u_n-u\right)d\mu \right]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & =o_n\left(1\right),\hfill \end{array}$$

(31)

which implies that $\parallel u_n{-u\parallel }_{p,V}^p=o_n\left(1\right)$, for $1<p<2$. Moreover, by using (27), we can obtain that $\parallel u_n{-u\parallel }_{q,V}^q=o_n\left(1\right)$. Consequently, $\parallel u_n-u\parallel =\parallel u_n{-u\parallel }_{p,V}+{\parallel u_n-u\parallel }_{q,V}=o_n\left(1\right)$. Thus, $u_n\to u$ in X as $n\to +\infty$. This completes the proof. □

Theorem 1. 

Let $G=(K^N,E,\mu ,\omega )$ be a finite weighted lattice graph. If $1<p<q<N$, and $max\left\{1,{\displaystyle \frac{q}{2}},{\displaystyle \frac{(N+\alpha )p}{2N}}\right\}<r\le {\displaystyle \frac{(N+\alpha )q}{2(N-q)}}$, then there exists a mountain pass solution of Equation (6).

Proof. 

By Lemma 4, the functional $J_{p,q}$ satisfies the geometric framework of the mountain pass theorem. Lemma 6 shows that $J_{p,q}$ satisfies the (PS) condition. We conclude by utilizing the mountain pass theorem [21] that $c={inf}_{\gamma \in {\mathsf{\Gamma }}_1}{max}_{t\in [0,1]}{J}_{p,q}\left(\gamma \left(t\right)\right)$ is a critical value of $J_{p,q}$, where

$${\mathsf{\Gamma }}_1:=\{\gamma \in C([0,1],X):\gamma \left(0\right)=0,\gamma \left(1\right)=e\}.$$

Thus, there exists some $u^{*}\in X$ such that $J_{p,q}\left(u^{*}\right)=c\ge \sigma >0$, and $u^{*}\ne 0$ is a nontrivial solution of (6). Therefore, we complete the proof of Theorem 1. □

3.2. Existence of Ground State Solution

In the following, we focus on the existence of the ground state solution of Equation (6). The Nehari manifold related to (6) is defined as ${\mathcal{N}}_{p,q}=\{u\in X\backslash \left\{0\right\}|J_{p,q}^{\prime }\left(u\right)\left(u\right)=0\}$, namely

$${\displaystyle {\mathcal{N}}_{p,q}:=\left\{u\in X\backslash {\left\{0\right\}:\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^q=\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu \right\}.}$$

(32)

Denote

$${\displaystyle m_{p,q}:=\underset{u\in {\mathcal{N}}_{p,q}}{inf}{J}_{p,q}\left(u\right).}$$

Lemma 7. 

Under the conditions of Theorem 1,

(i) 

there exists $\rho >0$, $\forall u\in {\mathcal{N}}_{p,q}$, and one has $\parallel u\parallel \ge \rho$;

(ii) 

$m_{p,q}>0$;

(iii) 

$\forall u\in X\backslash \left\{0\right\}$, and there exists a unique $t_u>0$ such that $t_{u}u\in {\mathcal{N}}_{p,q}$.

Proof. 

(i) For any $u\in {\mathcal{N}}_{p,q}$ with $\parallel u\parallel \le 1$, from (32) and (12), we deduce that

$${\displaystyle {\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^q=\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu \le C{\parallel u\parallel }^{2r}.}$$

(33)

Since ${\parallel u\parallel }_{p,V},{\parallel u\parallel }_{q,V}<1$, it follows from (13) that

$${\displaystyle {\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^q\ge {\parallel u\parallel }_{p,V}^q+{\parallel u\parallel }_{q,V}^q\ge c_q{(\parallel u\parallel }_{p,V}+{\parallel u\parallel }_{q,V}{)}^q=c_q{\parallel u\parallel }^q.}$$

(34)

Combining (33), (34) and $2r>q$, we conclude that there exists $1>\rho >0$ such that $\parallel u\parallel \ge \rho$, $\forall u\in {\mathcal{N}}_{p,q}$.

(ii) We first prove the following inequality:

$${\displaystyle {\parallel u\parallel }_{p,V}\le C{\parallel u\parallel }_{q,V}.}$$

(35)

In fact, by using the H$\ddot{\mathrm{o}}$lder inequality, one has

$$\begin{array}{cc}{\displaystyle \underset{K^N}{\int }{|\nabla u|}^{p}d\mu }\hfill & \le {\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{{\displaystyle \frac{p}{q}}}·{\left(\underset{K^N}{\int }{1}^{{\displaystyle \frac{q}{q-p}}}d\mu \right)}^{{\displaystyle \frac{q-p}{q}}}\hfill \\ \hfill & {\displaystyle ={\left(\sum _{x\in K^N}\mu \left(x\right)\right)}^{\frac{q-p}{q}}{\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{\frac{p}{q}}=C_4{\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{\frac{p}{q}},}\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}{\displaystyle \underset{K^N}{\int }V\left(x\right){\left|u\right|}^{p}d\mu }\hfill & =\underset{K^N}{\int }V{\left(x\right)}^{{\displaystyle \frac{q-p}{q}}}·V{\left(x\right)}^{{\displaystyle \frac{p}{q}}}{\left|u\right|}^{p}d\mu \hfill \\ \hfill & {\displaystyle \le {\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}}·{\left(\underset{K^N}{\int }V\left(x\right)d\mu \right)}^{\frac{q-p}{q}}\le {\left(V_m\sum _{x\in K^N}\mu \left(x\right)\right)}^{\frac{q-p}{q}}·{\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}}}\hfill \\ \hfill & {\displaystyle =C_5{\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}},}\hfill \end{array}$$

(37)

where $V_m={max}_{x\in K^N}V\left(x\right)<+\infty$. Thus,

$${\displaystyle \underset{K^N}{\int }{\left(\right|\nabla u|}^p+{V\left(x\right)\left|u\right|}^p)d\mu \le C_6\left[{\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{\frac{p}{q}}+{\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}}\right].}$$

(38)

From (13), we have

$${\displaystyle {(x+y)}^r\le C_r(x^r+y^r),x,y\ge 0,r>1,}$$

i.e.,

$${\displaystyle x+y\le C_7{(x^r+y^r)}^{\frac{1}{r}},x,y\ge 0,r>1.}$$

(39)

In (39), let

$${\displaystyle x={\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{\frac{p}{q}},y={\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}},r=\frac{q}{p}>1,}$$

and we can derive that

$${\displaystyle {\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu \right)}^{\frac{p}{q}}+{\left(\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}}\le C_7{\left(\underset{K^N}{\int }{|\nabla u|}^{q}d\mu +\underset{K^N}{\int }V\left(x\right){\left|u\right|}^{q}d\mu \right)}^{\frac{p}{q}}.}$$

(40)

Substituting (40) into (38), we deduce that (35) holds. It follows from (35) that

$${\displaystyle {(1+C)\parallel u\parallel }_{q,V}\ge {\parallel u\parallel }_{p,V}+{\parallel u\parallel }_{q,V}=\parallel u\parallel \ge \rho ,}$$

which implies that ${\parallel u\parallel }_{q,V}\ge C_8$. Thus, we have by (13) that

$$\begin{array}{cc}{\displaystyle {\parallel u\parallel }_{p,V}^p}\hfill & +{\parallel u\parallel }_{q,V}^q={\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^p·{\parallel u\parallel }_{q,V}^{q-p}\ge {\parallel u\parallel }_{p,V}^p+C_8^{q-p}{\parallel u\parallel }_{q,V}^p\hfill \\ \hfill & {\displaystyle \ge C_9{(\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^p)\ge C_{10}{(\parallel u\parallel }_{p,V}+{\parallel u\parallel }_{q,V}{)}^p=C_{10}{\parallel u\parallel }^p}\hfill \\ \hfill & {\displaystyle \ge C_{10}{\rho }^p.}\hfill \end{array}$$

(41)

In view of (32) and (41), we derive that

$$\begin{array}{l}{m}_{p,q}=\underset{u\in {\mathcal{N}}_{p,q}}{inf}\left(\frac{1}{p}{\parallel u\parallel }_{p,V}^p+\frac{1}{q}{\parallel u\parallel }_{q,V}^q-\frac{1}{2r}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}\frac{{\left|u\left(y\right)\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu \right)\\ =\underset{u\in {\mathcal{N}}_{p,q}}{inf}\left[\left(\frac{1}{p}-\frac{1}{2r}\right){\parallel u\parallel }_{p,V}^p+\left(\frac{1}{q}-\frac{1}{2r}\right){\parallel u\parallel }_{q,V}^q\right]\\ \ge \left(\frac{1}{q}-\frac{1}{2r}\right)\underset{u\in {\mathcal{N}}_{p,q}}{inf}{(\parallel u\parallel }_{p,V}^p+{\parallel u\parallel }_{q,V}^q)\ge \left(\frac{1}{q}-\frac{1}{2r}\right)C_{10}{\rho }^p>0.\end{array}$$

(iii) For each fixed $u\in X\backslash \left\{0\right\}$ and $\forall t\ge 0$, we set $w_u\left(t\right)=J_{p,q}\left(tu\right)$, namely

$${\displaystyle w_u\left(t\right)=\frac{t^p}{p}{\parallel u\parallel }_{p,V}^p+\frac{t^q}{q}{\parallel u\parallel }_{q,V}^q-\frac{t^{2r}}{2r}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu .}$$

(42)

Taking the derivative of $w_u\left(t\right)$, we obtain

$$\begin{array}{cc}{\displaystyle w_u^{\prime }\left(t\right)}\hfill & =t^{p-1}{\parallel u\parallel }_{p,V}^p+t^{q-1}{\parallel u\parallel }_{q,V}^q-t^{2r-1}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}{\displaystyle \frac{{\left|u\right|}^r}{d{(x,y)}^{N-\alpha }}}\right){\left|u\right|}^{r}d\mu \hfill \\ \hfill & {\displaystyle =t^{p-1}\left[{\parallel u\parallel }_{p,V}^p-t^{q-p}\left(t^{2r-q}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu -{\parallel u\parallel }_{q,V}^q\right)\right].}\hfill \end{array}$$

(43)

It is easy to see that the function

$${\displaystyle k_u\left(t\right)=t^{q-p}\left(t^{2r-q}\underset{K^N}{\int }\left(\sum _{y\in K^N,y\ne x}\frac{{\left|u\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu -{\parallel u\parallel }_{q,V}^q\right)}$$

(44)

is strictly increasing on $(t_u^{*},+\infty )$, and $k_u\left(t_u^{*}\right)=0$, where

$${\displaystyle t_u^{*}={\left(\frac{{\parallel u\parallel }_{q,V}^q}{\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}\frac{{\left|u\right|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u\right|}^{r}d\mu }\right)}^{\frac{1}{2r-q}}.}$$

It follows from (43) and (44) that there exists a unique $t_u>t_u^{*}$ such that $w_u^{\prime }\left(t\right)>0$ on $(t_u^{*},t_u)$ and $w_u^{\prime }\left(t\right)<0$ on $(t_u,+\infty )$. Moreover, if $t\in (0,t_u^{*}]$, then $w_u^{\prime }\left(t\right)>0$. Thus, ${max}_{t\ge 0}{w}_u\left(t\right)=w_u\left(t_u\right)$ and $w_u^{\prime }\left(t_u\right)=0$, i.e., $t_{u}u\in {\mathcal{N}}_{p,q}$. □

Lemma 8. 

Set maps $t:X\backslash \left\{0\right\}\to (0,+\infty ):u\mapsto t\left(u\right):=t_u$, and $\lambda :X\backslash \left\{0\right\}\to {\mathcal{N}}_{p,q}:u\mapsto \lambda \left(u\right):=t_{u}u$. Under the conditions of Theorem 1,

(i) 

t and λ are continuous;

(ii) 

${\lambda |}_S$ (S represents the unit sphere in X) is a homeomorphism between S and ${\mathcal{N}}_{p,q}$.

Proof. 

Assume that $u_n\to u$ in $X\backslash \left\{0\right\}$ as $n\to +\infty$, and it suffices to show that $t_{u_n}\to t_u$. Set $t_{u_n}=t_n$; then, $\lambda \left(u_n\right)=t_n{u}_n$. Since $\lambda \left(su\right)=\lambda \left(u\right)$ for any $s>0$, we may assume that $\parallel u_n\parallel =1$. In the following, we will prove that there exist $\delta >0$, $C_S>0$, such that $\delta \le t_n\le C_S$, $\forall n\in \mathbb{N}$. Indeed, $t_n$ is the unique root for

$${\displaystyle w_{u_n}^{\prime }\left(t_n\right)=t_n^{p-1}{\theta }_{u_n}\left(t_n\right)=0,}$$

where

$$\begin{array}{l}{\theta }_{u_n}\left(t\right)=\parallel u_n{\parallel }_{p,V}^p-t^{2r-p}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}\frac{|u_n{|}^r}{d{(x,y)}^{N-\alpha }}\right)|u_n{|}^{r}d\mu +t^{q-p}{\parallel u_n\parallel }_{q,V}^q\\ \ge \parallel u_n{\parallel }_{p,V}^p-t^{2r-p}\underset{K^N}{\int }\left({\displaystyle \sum _{y\in K^N,y\ne x}}\frac{|u_n{|}^r}{d{(x,y)}^{N-\alpha }}\right){\left|u_n\right|}^{r}d\mu \\ \ge \parallel u_n{\parallel }_{p,V}^p-C{t}^{2r-p}{\parallel u_n\parallel }^{2r}\\ =\parallel u_n{\parallel }_{p,V}^p-C{t}^{2r-p}.\end{array}$$

Taking $\delta >0$ small enough, one has

$${\displaystyle {\theta }_{u_n}\left(t\right)>0,\forall t\in (0,\delta ).}$$

Hence,

$${\displaystyle w_{u_n}^{\prime }\left(t\right)=t^{p-1}{\theta }_{u_n}\left(t\right)>0,\forall t\in (0,\delta ),}$$

and then $t_n>\delta >0$. On the other hand, we will prove that there exists $C_S>0$ such that $t_n\le C_S$. We show this by contradiction. Assume that $t_n\to +\infty$, $n\to +\infty$. Observe that

$${\displaystyle w_{u_n}\left(t_n\right)=\underset{t\ge 0}{max}{w}_{u_n}\left(t\right)=\underset{t\ge 0}{max}{J}_{p,q}\left(t{u}_n\right)>0.}$$

Nevertheless, by (15), we can easily find that $w_{u_n}\left(t_n\right)=J_{p,q}\left(t_n{u}_n\right)\to -\infty$ as $n\to +\infty$, which is a contradiction. Thus, we may assume that $t_n\to \overline{t}>0$ after passing to the subsequence. Then, $\lambda \left(u_n\right)=t_n{u}_n\to \overline{t}u$ as $n\to +\infty$. Thus, we simply need to demonstrate that $\overline{t}=t_u$. Since ${\mathcal{N}}_{p,q}$ is closed and $t_n{u}_n\in {\mathcal{N}}_{p,q}$, $\overline{t}u\in {\mathcal{N}}_{p,q}$. According to the uniqueness of $t_u$, one has $\overline{t}=t_u$ and $\lambda \left(u_n\right)\to \overline{t}u=t_{u}u=\lambda \left(u\right)$. Hence, the map t and $\lambda$ are continuous.

(ii) Let $u_i\in S$, $\lambda \left(u_i\right)=t_i{u}_i$, $i=1,2$. If $\lambda \left(u_1\right)=\lambda \left(u_2\right)$, i.e., $t_1{u}_1=t_2{u}_2$, one has $t_1=t_2>\delta >0$ after taking the norm on both sides. Thus, we have $u_1=u_2$ and ${\lambda |}_S$ is injective. For each $u\in {\mathcal{N}}_{p,q}$, let $\overline{u}={\displaystyle \frac{u}{\parallel u\parallel }}$; then, $\overline{u}\in S$. Owing to $u=\parallel u\parallel \overline{u}$ and $t_{\overline{u}}$ being unique, we have $t_{\overline{u}}=\parallel u\parallel$. Thus, $\lambda \left(\overline{u}\right)=t_{\overline{u}}\overline{u}=u={\mathcal{N}}_{p,q}$, i.e., ${\lambda |}_S$ is surjective. Therefore, ${\lambda |}_S:S\to {\mathcal{N}}_{p,q}$ is a homeomorphism. We complete the proof of Lemma 8. □

Define

$${\displaystyle c_1:=\underset{u\in X\backslash \left\{0\right\}}{inf}\underset{t\ge 0}{max}{J}_{p,q}\left(tu\right),c_2:=\underset{\gamma \in {\mathsf{\Gamma }}_2}{inf}\underset{t\in [0,1]}{max}{J}_{p,q}\left(\gamma \left(t\right)\right),}$$

(45)

where

$${\displaystyle {\mathsf{\Gamma }}_2:=\{\gamma \in C([0,1],X):\gamma \left(0\right)=0,J_{p,q}\left(\gamma \left(1\right)\right)<0\}.}$$

In what follows, we will prove that $c_1$ and $c_2$ are equal.

Lemma 9. 

$c_1=c_2=m_{p,q}>0$.

Proof. 

Firstly, we prove $c_1=m_{p,q}$. From Lemma 7, we understand that a unique $t_u>0$ exists for which

$${\displaystyle \underset{t\ge 0}{max}{w}_u\left(t\right)=\underset{t\ge 0}{max}{J}_{p,q}\left(tu\right)=w_u\left(t_u\right)=J_{p,q}\left(t_{u}u\right),}$$

where $w_u\left(t\right)$ is defined by (42). Thus, one has

$${\displaystyle c_1=\underset{u\in X\backslash \left\{0\right\}}{inf}\underset{t\ge 0}{max}{J}_{p,q}\left(tu\right)=\underset{u\in X\backslash \left\{0\right\}}{inf}{J}_{p,q}\left(t_{u}u\right)=\underset{u\in {\mathcal{N}}_{p,q}}{inf}{J}_{p,q}\left(u\right)=m_{p,q}.}$$

(46)

Secondly, we prove $c_1\ge c_2$. It follows from (15) that there is a large enough $t_0>0$ such that $w_u\left(t_0\right)<0$ for each $u\in X\backslash \left\{0\right\}$. Define ${\gamma }_0:[0,1]\to X:t\mapsto t_{0}tu$. Since ${\gamma }_0\left(0\right)=0$, $J_{p,q}\left({\gamma }_0\left(1\right)\right)<0$, which implies that ${\gamma }_0\in {\mathsf{\Gamma }}_2$. Thus, for each $u\in X\backslash \left\{0\right\}$, we have

$${\displaystyle \underset{t\ge 0}{max}{J}_{p,q}\left(tu\right)\ge \underset{t\in [0,1]}{max}{J}_{p,q}\left(t_{0}tu\right)=\underset{t\in [0,1]}{max}{J}_{p,q}\left({\gamma }_0\left(t\right)\right)\ge \underset{\gamma \in {\mathsf{\Gamma }}_2}{inf}\underset{t\in [0,1]}{max}{J}_{p,q}\left(\gamma \left(t\right)\right),}$$

which implies

$${\displaystyle c_1=\underset{u\in X\backslash \left\{0\right\}}{inf}\underset{t\ge 0}{max}{J}_{p,q}\left(tu\right)\ge \underset{\gamma \in {\mathsf{\Gamma }}_2}{inf}\underset{t\in [0,1]}{max}{J}_{p,q}\left(\gamma \left(t\right)\right)=c_2.}$$

(47)

Finally, we prove $c_2\ge m_{p,q}$. By Lemma 7, for each $u\in X\backslash \left\{0\right\}$, there exists a unique $t_u>0$ such that $t_{u}u\in {\mathcal{N}}_{p,q}$. We can subsequently split X into two components, namely $X=X_1\cup X_2$, where $X_1=\{u\in X:t_u\ge 1\}$ and $X_2=\{u\in X:t_u<1\}$. We now assert that every $\gamma \in {\mathsf{\Gamma }}_2$ must cross ${\mathcal{N}}_{p,q}$. Indeed, if t is small enough, we can easily find that $\gamma \left(t\right)$ and 0 belong to the component $X_1$. Consider the function $w_{\gamma \left(1\right)}\left(t\right)=J_{p,q}\left(t\gamma \left(1\right)\right)$, $\forall t\in [0,+\infty )$, and we have $w_{\gamma \left(1\right)}\left(0\right)=0$ and $w_{\gamma \left(1\right)}\left(1\right)<0$. From (12), it holds that

$${\displaystyle w_{\gamma \left(1\right)}\left(t\right)\ge \frac{t^p}{p}{\parallel \gamma \left(1\right)\parallel }_{p,V}^p+\frac{t^q}{q}{\parallel \gamma \left(1\right)\parallel }_{q,V}^q-C_{11}\frac{t^{2r}}{2r}{\parallel \gamma \left(1\right)\parallel }^{2r},}$$

which suggests that $w_{\gamma \left(1\right)}\left(t\right)>0$ for small enough $t>0$. Thus, there exist $t_{\gamma \left(1\right)}\in (0,1)$ such that ${max}_{t\ge 0}{w}_{\gamma \left(1\right)}\left(t\right)=J_{p,q}\left(t_{\gamma \left(1\right)}\gamma \left(1\right)\right)$. Thus, $t_{\gamma \left(1\right)}<1$ and $\gamma \left(1\right)\in X_2$. Furthermore, according to Lemma 8, the map $u\mapsto t_u$ is continuous, which means that every $\gamma \in {\mathsf{\Gamma }}_2$ has to cross ${\mathcal{N}}_{p,q}$. Hence, $\forall \gamma \in {\mathsf{\Gamma }}_2$, and there exists $t_0\in (0,1)$ such that $\gamma \left(t_0\right)\in {\mathcal{N}}_{p,q}$. Thus, we have

$${\displaystyle \underset{u\in {\mathcal{N}}_{p,q}}{inf}{J}_{p,q}\left(u\right)\le J_{p,q}\left(\gamma \left(t_0\right)\right)\le \underset{t\in [0,1]}{max}{J}_{p,q}\left(\gamma \left(t\right)\right),}$$

which implies that

$${\displaystyle m_{p,q}=\underset{u\in {\mathcal{N}}_{p,q}}{inf}{J}_{p,q}\left(u\right)\le \underset{\gamma \in {\mathsf{\Gamma }}_2}{inf}\underset{t\in [0,1]}{max}{J}_{p,q}\left(\gamma \left(t\right)\right)=c_2.}$$

(48)

Combining (46), (47), (48) and Lemma 7, we find that $c_1=c_2=m_{p,q}>0$ holds, which gives the desired result. □

Theorem 2. 

Let $G=(K^N,E,\mu ,\omega )$ be a finite weighted lattice graph. If $1<p<q<N$, and $max\left\{1,{\displaystyle \frac{q}{2}},{\displaystyle \frac{(N+\alpha )p}{2N}}\right\}<r\le {\displaystyle \frac{(N+\alpha )q}{2(N-q)}}$, then Equation (6) admits a ground state solution.

Proof. 

From Lemmas 4 and 6, $J_{p,q}$ satisfies the mountain pass geometry and the ${\left(PS\right)}_{c_2}$ condition, with $c_2$ defined in (45). According to the mountain pass theorem of [25] and Lemma 9, there exists a solution u such that $J_{p,q}\left(u\right)=c_2=m_{p,q}>0$, which completes the proof. □

Remark 2. 

Theorem 2 generalizes the main result of Liu and Zhang [18] in two aspects: from phase operator $-{\Delta }_p$ to double phasic operator $-{\Delta }_p-{\Delta }_q$ and from finite lattice graphs to finite weighted lattice graphs.

4. Conclusions

In this work, we studied the nonlinear Choquard equations with $(p,q)$-Laplacians on finite weighted lattice graphs. By using the mountain pass theorem and the method of the Nehari manifold, the existence of a mountain pass solution and ground state solution has been established, respectively. To the best of our knowledge, there are no existing research results for this type of equation on weighted lattice graphs. Our research findings enrich the field of study regarding differential equations on graphs.

In the future, we will explore the existence of ground state solutions for the fractional Kirchhoff equation with critical exponents on locally finite weighted graphs, in addition to the existence of normalized solutions for the p-biharmonic Choquard equation on weighted graphs.