Infinitely Many Solutions for Partial Discrete Kirchhoff Type Problems Involving p-Laplacian

Abstract

In this paper, the existence of infinitely many solutions for the partial discrete Kirchhoff-type problems involving p-Laplacian is proven by exploiting the critical point theory for the first time. Moreover, by using the strong maximum principle, we acquire some sufficient conditions for the presence of infinitely many positive solutions to the boundary value problems. Our major outcomes are explained with one example.

1. Introduction

For $\gamma \in Z$, define $Z(1,\gamma )=\{1,2,\cdots ,\gamma \}$. Our focus is on a discrete Kirchhoff-type problem shown below, discrete Kirchhoff-type problems is DKP for short

$$\begin{array}{c}{-(a+b\parallel u\parallel }^p\left)\right({\mathrm{\Delta }}_1({\varphi }_p({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))=\lambda f((\kappa ,\nu ),u(\kappa ,\nu )),(\kappa ,\nu )\in [1,\gamma ]\times [1,\tau ],\hfill \end{array}$$

with boundary conditions

$$\begin{array}{c}\hfill u(\kappa ,0)=u(\kappa ,\tau +1)=0,\kappa \in [0,\gamma +1],\\ \hfill u(0,\nu )=u(\gamma +1,\nu )=0,\nu \in [0,\tau +1],\end{array}$$

(1)

where a, b, $\gamma$ and $\tau$ are four given positive constants, $\lambda$ denotes a positive real variable, ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are the forward difference operator defined by ${\mathrm{\Delta }}_{1}u(\kappa ,\nu )=u(\kappa +1,\nu )-u(\kappa ,\nu )$ and ${\mathrm{\Delta }}_{2}u(\kappa ,\nu )=u(\kappa ,\nu +1)-u(\kappa ,\nu )$, ${\mathrm{\Delta }}_1^{2}u(\kappa ,\nu )={\mathrm{\Delta }}_1\left({\mathrm{\Delta }}_{1}u(\kappa ,\nu )\right)$ and ${\mathrm{\Delta }}_2^{2}u(\kappa ,\nu )={\mathrm{\Delta }}_2\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu )\right)$, ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $p>1$, and $f\left(\right(\kappa ,\nu ),u)$ is continuous in u for all $(\kappa ,\nu )\in [0,\gamma +1]\times [0,\tau +1]$.

Difference equations have been widely used in diverse domains [1,2,3,4]. Important tools for the study of difference equations include fixed point methods [5,6]. In 2003, for the first time, Yu and Guo [7] investigated a class of difference equations. Since then, numerous researchers have studied difference equations and achieve a lot of findings, including the findings of periodic solutions [7,8,9,10], homoclinic solutions [11,12,13,14,15,16,17,18,19,20,21,22], and boundary value problems [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37].

It is worth noting that the aforementioned difference equations only have one variable. In contrast, there have been fewer studies on difference equations involving two variables, also known as partial difference equations. Such equations have recently become popular in a variety of fields. However, the attention of significant mathematical scholars has been paid to the boundary value problem for partial difference equations [38,39,40], and other meaningful results [41,42,43,44,45,46]. In recent years, an increasing number of researchers have focused on the Schrödinger equations, especially for the DKP. We refer to [47,48] and their references for the relevant studies.

In 2022, Du and Zhou [40] explored the following problem

$${\mathrm{\Delta }}_1\left({\varphi }_c\left({\mathrm{\Delta }}_{1}s(\kappa -1,\nu )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_c\left({\mathrm{\Delta }}_{2}s(\kappa ,\nu -1)\right)\right)+\lambda f((\kappa ,\nu ),s(\kappa ,\nu ))=0,(\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau ),$$

(2)

with boundary conditions (1), and proved the existence of multiple solutions of (2).

In comparison with the findings of the partial difference equations, the discrete Schrödinger equations of the Kirchhoff type have been less investigated. In actuality, the discrete Kirchhoff-type problem has strong theoretical significance and application value [3,4]. The authors in [31] considered infinitely many solutions of the difference equations, and the problem only contains one discrete variable. Different from [31], in this paper, we consider the partial difference equations with a Kirchhoff-type problem and the equations have two discrete variables. In [32], the author considered the three solutions of the partial difference equations, and the main method was to refer to Theorem 2.1 in [33]. Thus, the method and the results are different from those in the previous literature. Owing to the above reasons, we attempt to study the existence of infinitely many solutions for the partial discrete Kirchhoff-type problems involving p-Laplacian for the first time. The contributions and novelty of this paper are summarized as follows:

(1)

This paper is the first time to consider the infinitely many solutions of the partial discrete Kirchhoff-type problems involving p-Laplacian, which are more complex to deal with.

(2)

The difficulty to be overcome in this paper is the estimation of $\parallel u\parallel$ in Theorem 1.

(3)

To prove the existence of infinitely many solutions of the partial discrete Kirchhoff-type problems involving p-Laplacian, we use critical point theory. Further, by virtue of the strong maximum principle we have established, we acquire some sufficient conditions for the presence of infinitely many positive solutions to the boundary value problems.

(4)

We give one example to illustrate our conclusion.

We arrange the reminder of this paper as shown below. In Section 2, we give preliminaries. In Section 3, we present some results of this paper. In Section 4, our major findings are explained using one example.

2. Preliminaries

As the initial step of this section, the DKP-related variational framework is constructed. Suppose that there is the following $\gamma \tau$-dimensional Banach space

$U=\{u:\mathbb{Z}(0,\gamma +1)\times \mathbb{Z}(0,\tau +1)\to \mathbb{R}:u(\kappa ,0)=u(\kappa ,\tau +1)=0,\forall \kappa \in \mathbb{Z}(0,\gamma +1)$ and $u(0,\nu )=u(\gamma +1,\nu )=0,\forall \nu \in \mathbb{Z}(0,\tau +1)\}$,

$$\parallel u\parallel ={\left(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma +1}{\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)}^p+\sum _{\kappa =1}^{\gamma }\sum _{\nu =1}^{\tau +1}{\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)}^p\right)}^{\frac{1}{p}},\forall u\in U.$$

Put

$$\mathrm{\Phi }\left(u\right)=\frac{a}{p}{\parallel u\parallel }^p+\frac{b}{2p}{\parallel u\parallel }^{2p},\mathsf{\Psi }\left(u\right)=\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }F((\kappa ,\nu ),u(\kappa ,\nu )),$$

for every $u\in U$, where $F((\kappa ,\nu ),u)={\int }_0^{u}f((\kappa ,\nu ),s)ds$ is the main role of $f\left(\right(\kappa ,\nu ),u)$ with $F\left(\right(\kappa ,\nu ),0)=0$ for each $\left(\right(\kappa ,\nu ),u)\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau )\times \mathbb{R}$. Let

$$I_{\lambda }\left(u\right)=\mathrm{\Phi }\left(u\right)-\lambda \mathsf{\Psi }\left(u\right),$$

for any $u\in U$. By careful calculation, we have

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}^{\prime }\left(u\right)\left(v\right)& =\underset{t\to 0}{lim}\frac{\mathrm{\Phi }(u+tv)-\mathrm{\Phi }\left(u\right)}{t}\hfill \\ \hfill & =-(a+b(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma +1}|{\mathrm{\Delta }}_1{u(\kappa -1,\nu )|}^p+\sum _{\kappa =1}^{\gamma }\sum _{\nu =1}^{\tau +1}|{\mathrm{\Delta }}_2{u(\kappa ,\nu -1)|}^p\left)\right)(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }{\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)\hfill \\ \hfill & +\sum _{\kappa =1}^{\gamma }\sum _{\nu =1}^{\tau }{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))v(\kappa ,\nu )\hfill \\ \hfill & ={(a+b\parallel u\parallel }^p)(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }{\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)+\sum _{\kappa =1}^{\gamma }\sum _{\nu =1}^{\tau }{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))v(\kappa ,\nu )\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\mathsf{\Psi }}^{\prime }\left(u\right)\left(v\right)=\underset{t\to 0}{lim}\frac{\mathsf{\Psi }(u+tv)-\mathsf{\Psi }\left(u\right)}{t}=\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }f((\kappa ,\nu ),u(\kappa ,\nu ))v(\kappa ,\nu ),\end{array}$$

for $\forall u,v\in U$. Obviously, for any $u,v\in U$,

$$\begin{array}{cc}\hfill {(\mathrm{\Phi }-\lambda \mathsf{\Psi })}^{\prime }\left(u\right)\left(v\right)& =-\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }\left[\right(a+{b\parallel u\parallel }^p\left)\right({\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right)\hfill \\ \hfill & +\lambda f\left(\right(\kappa ,\nu ),u(\kappa ,\nu \left)\right)\left)\right]v(\kappa ,\nu ),\forall v(\kappa ,\nu )\in U,\hfill \end{array}$$

(3)

is equivalent to

$${-(a+b\parallel u\parallel }^p)({\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))=\lambda f((\kappa ,\nu ),u(\kappa ,\nu )),$$

Apparently, $\mathrm{\Phi },\mathsf{\Psi }\in C^1(U,R)$, we change the existence of a solution for the problem DKP into the existence of a critical point of $\mathrm{\Phi }-\lambda \mathsf{\Psi }$ on U.

Let X be a reflexive real Banach space and let $I_{\lambda }:X\to R$ be a function which satisfies the following structure hypothesis:

$\left(\mathrm{\Lambda }\right)I_{\lambda }\left(u\right):=\mathrm{\Phi }\left(u\right)-\lambda \mathsf{\Psi }\left(u\right)$ for all $u\in X$, where $\mathrm{\Phi },\mathsf{\Psi }:X\to R$ are two functions of class $C^1$ on X with $\mathrm{\Phi }$ coercive, i.e., $\underset{\parallel u\parallel \to +\infty }{lim}\mathrm{\Phi }\left(u\right)=+\infty$, and $\lambda$ is a real positive parameter.

If $\underset{X}{inf}\mathrm{\Phi }<r$, let

$$\phi \left(r\right):=\underset{u\in {\mathrm{\Phi }}^{-1}(-\infty ,r)}{inf}\frac{\left(\underset{v\in {\mathrm{\Phi }}^{-1}(-\infty ,r)}{sup}\mathsf{\Psi }\left(v\right)\right)-\mathsf{\Psi }\left(u\right)}{r-\mathrm{\Phi }\left(u\right)},$$

$$\delta :=\underset{r\to {\left({inf}_X\mathrm{\Phi }\right)}^{+}}{lim\; inf}\phi \left(r\right),\gamma :=\underset{r\to +\infty }{lim\; inf}\phi \left(r\right).$$

Obviously, $\delta \ge 0$ and $\gamma \ge 0$. When $\delta =0$ (or $\gamma =0$), in the sequel, we agree to read $\frac{1}{\delta }$ (or $\frac{1}{\gamma }$ ) as $+\infty$. The following lemma comes from Theorem 2.1 of [49] and will be used to investigate problem DKP.

Lemma 1. 

Assume that the condition $\left(\mathsf{\Lambda }\right)$ holds. We have

$\left(a\right)$

If $\gamma <+\infty$, then, for each $\lambda \in (0,\frac{1}{\gamma })$, the following alternative holds: either

$\left(a_1\right)$

$I_{\lambda }$ possesses a global minimum, or

$\left(a_2\right)$

there is a sequence $\left\{u_n\right\}$ of critical points (local minima) of $I_{\lambda }$ such that $\underset{n\to +\infty }{lim}\mathrm{\Phi }\left(u_n\right)=+\infty$.

The strong maximum principle shown below is required for acquiring the positive solution to problem DKP.

Lemma 2. 

Suppose there exists u: $\mathbb{Z}(0,\gamma +1)\times \mathbb{Z}(0,\tau +1)\to \mathbb{R}$ such that

$${u(\kappa ,\nu )>0or-(a+b\parallel u\parallel }^p)({\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))\ge 0,$$

(4)

for all $(\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau )$. Then, either $u>0$ or $u\equiv 0$ for all $(\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau )$.

Proof. 

Let $ϵ\in \mathbb{Z}(1,\gamma ),\varsigma \in \mathbb{Z}(1,\tau )$ and $u(ϵ,\varsigma )=min\left\{u\right(\kappa ,\nu ):\kappa \in \mathbb{Z}(1,\gamma ),\nu \in \mathbb{Z}(1,\tau \left)\right\}$.

If $u(ϵ,\varsigma )>0$, then it is obvious that $u(\kappa ,\nu )>0$ for all $\kappa \in \mathbb{Z}(1,\gamma ),\nu \in \mathbb{Z}(1,\tau )$.

If $u(ϵ,\varsigma )\le 0$, due to $a,b>0$, then we have

$${\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(ϵ,\varsigma -1)\right)\right)\le 0.$$

(5)

$u(ϵ,\varsigma )=min\left\{u\right(\kappa ,\nu ):\kappa \in \mathbb{Z}(1,\gamma +1),\nu \in \mathbb{Z}(1,\tau +1\left)\right\}$, since ${\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )=u(ϵ,\varsigma )-u(ϵ-1,\varsigma )\le 0$, ${\mathrm{\Delta }}_{2}u(ϵ,\varsigma -1)=u(ϵ,\varsigma )-u(ϵ,\varsigma -1)\le 0$, and ${\mathrm{\Delta }}_{1}u(ϵ,\varsigma )=u(ϵ+1,\varsigma )-u(ϵ,\varsigma )\ge 0$, ${\mathrm{\Delta }}_{2}u(ϵ,\varsigma )=u(ϵ,\varsigma +1)-u(ϵ,\varsigma )\ge 0$. Since ${\varphi }_p\left(s\right)$ is increasing in s, and ${\varphi }_p\left(0\right)=0$, one has

$${\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)\le 0\le {\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ,\varsigma )\right),$$

which implies that ${\mathrm{\Delta }}_1\left[{\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)\right]\ge 0$. Similarly, ${\mathrm{\Delta }}_2\left[{\varphi }_p\left({\mathrm{\Delta }}_{2}u(ϵ,\varsigma -1)\right)\right]\ge 0$.

Thus,

$${\mathrm{\Delta }}_1\left[{\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)\right]+{\mathrm{\Delta }}_2\left[{\varphi }_p\left({\mathrm{\Delta }}_{2}u(ϵ,\varsigma -1)\right)\right]\ge 0.$$

(6)

Combining (5) with (6), we have ${\mathrm{\Delta }}_1\left[{\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)\right]+{\mathrm{\Delta }}_2\left[{\varphi }_p\left({\mathrm{\Delta }}_{2}u(ϵ,\varsigma -1)\right)\right]=0$.

So ${\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ,\varsigma )\right)={\varphi }_p\left({\mathrm{\Delta }}_{1}u(ϵ-1,\varsigma )\right)=0$. That is $u(ϵ+1,\varsigma )=u(ϵ-1,\varsigma )=u(ϵ,\varsigma )$. If $ϵ+1=\gamma +1$, we have $u(ϵ,\varsigma )=0$. Otherwise, $(ϵ+1)\in \mathbb{Z}(1,\gamma )$. Replacing $ϵ$ by $ϵ+1$, we get $u(ϵ+2,\varsigma )=u(ϵ+1,\varsigma )$. Continuing this process $(\gamma +1-ϵ)$ times, we have $u(ϵ,\varsigma )=u(ϵ+1,\varsigma )=u(ϵ+2,\varsigma )=\cdots =u(\gamma +1,\varsigma )=0$. Similarly, we have $u(ϵ,\varsigma )=u(ϵ-1,\varsigma )=u(ϵ-2,\varsigma )=\cdots =u(0,\varsigma )=0$. In the same way, we have $u\equiv 0$. Thus, the proof is complete. □

3. Main Results

Put

$$\begin{array}{c}\hfill B=\underset{\epsilon \to +\infty }{lim\; sup}\frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}F((\kappa ,\nu ),\epsilon )}{{\epsilon }^{2p}}.\end{array}$$

Theorem 1. 

Suppose two real sequences $\left\{a_t\right\}$ and $\left\{b_t\right\}$ exist, with $b_t>0$ and $\underset{t\to +\infty }{lim}{b}_t=+\infty$, such that

$$\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}>\frac{2}{p}(\gamma +\tau )a_t^p(a+b(\gamma +\tau )a_t^p),$$

(7)

for $t\in \mathbb{Z}\left(1\right)$, and

$$A=\underset{t\to \infty }{lim\; inf}\frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}\underset{\left|\epsilon \right|\le b_t}{max}F((\kappa ,\nu ),\epsilon )-{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}F((\kappa ,\nu ),a_t)}{\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}-\frac{2}{p}(\gamma +\tau )a_t^p(a+b(\gamma +\tau )a_t^p)}<\frac{pB}{2b{(\gamma +\tau )}^2}.$$

(8)

Then, for each $\lambda \in (\frac{2b{(\gamma +\tau )}^2}{pB},\frac{1}{A})$, problem DKP admits an unbounded sequence of solutions.

Proof. 

We validate our conclusion by Lemma 1. First of all, $\left(\mathsf{\Lambda }\right)$ is obviously satisfied. Put

$$r_t=\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}.$$

Due to proposition 1 of [39], if

$$\parallel u\parallel \le \frac{4}{{(\gamma +\tau +2)}^{\frac{p-1}{p}}}{\left[{({\left(\frac{a{(\gamma +\tau +2)}^{p-1}}{b{4}^p}\right)}^2+\frac{2p{(\gamma +\tau +2)}^{2p-2}{r}_t}{b{4}^{2p}})}^{\frac{1}{2}}-\frac{a{(\gamma +\tau +2)}^{p-1}}{b{4}^p}\right]}^{\frac{1}{p}},$$

then $\left|u(\kappa ,\nu )\right|\le b_t$ for every $(\kappa ,\nu )\in [1,\gamma ]\times [1,\tau ]$.

According to the definition of $\phi$,

$$\phi \left(r_t\right)\le \underset{u\in {\mathrm{\Phi }}^{-1}(-\infty ,r_t)}{inf}\frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}\underset{\left|\epsilon \right|\le b_t}{max}F((\kappa ,\nu ),\epsilon )-{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}F((\kappa ,\nu ),a_t)}{\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}-\mathrm{\Phi }\left(u\right)}.$$

For each $t\in \mathbb{Z}\left(1\right)$, let $w_t\in U$, define

$$w_t(\kappa ,\nu )=\left\{\begin{array}{cc}{a}_t,\hfill & (\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau ),\hfill \\ 0,\hfill & \kappa =0,\nu \in \mathbb{Z}(0,\tau +1),\mathrm{or},\kappa =\gamma +1,\nu \in \mathbb{Z}(0,\tau +1),\hfill \\ 0,\hfill & \nu =0,\kappa \in \mathbb{Z}(0,\gamma +1),\mathrm{or},\nu =\tau +1,\kappa \in \mathbb{Z}(0,\gamma +1).\hfill \end{array}\right.$$

Then

$$\mathrm{\Phi }\left(w_t\right)=\frac{2}{p}(\gamma +\tau )a_t^p(a+b(\gamma +\tau )a_t^p)<r_t$$

by using (7). Therefore,

$$\begin{array}{cc}\hfill \phi \left(r_t\right)& \le \frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}\underset{\left|\epsilon \right|\le b_t}{max}F((\kappa ,\nu ),\epsilon )-{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}F((\kappa ,\nu ),\left(w_t(\kappa ,\nu )\right))}{\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}-\mathrm{\Phi }\left(w_t\right)}\hfill \\ \hfill & =\frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}\underset{\left|\epsilon \right|\le b_t}{max}F((\kappa ,\nu ),\epsilon )-{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}F((\kappa ,\nu ),a_t)}{\frac{2a{4}^p{b}_t^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{b}_t^{2p}}{2p{(r+\tau +2)}^{2p-2}}-\frac{2}{p}(\gamma +\tau )a_t^p(a+b(\gamma +\tau )a_t^p)}.\hfill \end{array}$$

(9)

Therefore, by (8), we have

$$\gamma \le \underset{t\to +\infty }{lim\; inf}\phi \left(r_t\right)\le A<+\infty .$$

In order to get the conclusion $\left(a_2\right)$ of Lemma 1, we need to prove that $I_{\lambda }$ is unbounded from below. Therefore, the two cases $B=+\infty$ and $B<+\infty$ are taken into consideration.

In the case in which $B=+\infty$, and with $L>0$, we have $L>\frac{2b{(\gamma +\tau )}^2}{p\lambda }$. Assume $\left\{c_t\right\}$ is a sequence of positive numbers, and $\underset{t\to +\infty }{lim}{c}_t=+\infty$, which enables

$$\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }F((\kappa ,\nu ),c_t)\ge L·c_t^{2p},\mathrm{for}t\in \mathbb{Z}\left(1\right).$$

Setting a sequence $\left\{{\eta }_t\right\}$ in U with

$${\eta }_t(\kappa ,\nu )=\left\{\begin{array}{cc}{c}_t,\hfill & (\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau ),\hfill \\ 0,\hfill & \kappa =0,\nu \in \mathbb{Z}(0,\tau +1),\mathrm{or},\kappa =\gamma +1,\nu \in \mathbb{Z}(0,\tau +1),\hfill \\ 0,\hfill & \nu =0,\kappa \in \mathbb{Z}(0,\gamma +1),\mathrm{or},\nu =\tau +1,\kappa \in \mathbb{Z}(0,\gamma +1),\hfill \end{array}\right.$$

we get

$$\begin{array}{cc}\hfill I_{\lambda }\left({\eta }_t\right)& =\frac{2}{p}(\gamma +\tau )c_t^p(a+b(\gamma +\tau )c_t^p)-\lambda \left(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }F((\kappa ,\nu ),c_t)\right)\hfill \\ \hfill & \le \frac{2a}{p}(\gamma +\tau )c_t^p+(\frac{2b}{p}{(\gamma +\tau )}^2-\lambda L)c_t^{2p}\hfill \\ \hfill & <-\infty .\hfill \end{array}$$

(10)

If $B<+\infty$, since $\lambda >\frac{2b{(\gamma +\tau )}^2}{pB}$, ${\epsilon }_0>0$ can be selected. Thus, we have

$$\frac{2b}{p}{(\gamma +\tau )}^2-\lambda (B-{\epsilon }_0)<0.$$

Then, we propose a sequence $\left\{c_t\right\}$ meeting $\underset{t\to +\infty }{lim}{c}_t=+\infty$ and

$$(B-{\epsilon }_0)c_t^{2p}\le \sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }F((\kappa ,\nu ),c_t)\le (B+{\epsilon }_0)c_t^{2p}.$$

When the sequence $\left\{{\eta }_t\right\}$ in U is identical to the case in which $B=+\infty$. Thus, we have

$$\begin{array}{cc}\hfill I_{\lambda }\left({\eta }_t\right)& =\frac{2}{p}(\gamma +\tau )c_t^p(a+b(\gamma +\tau )c_t^p)-\lambda \left(\sum _{\nu =1}^{\tau }\sum _{\kappa =1}^{\gamma }F((\kappa ,\nu ),c_t)\right)\hfill \\ \hfill & \le \frac{2a}{p}(\gamma +\tau )c_t^p+(\frac{2b}{p}{(\gamma +\tau )}^2-\lambda (B-{\epsilon }_0))c_t^{2p}\hfill \\ \hfill & <-\infty .\hfill \end{array}$$

(11)

Combining the above two cases, we see that $I_{\lambda }$ is unbounded from below and by Lemma 1, we consider that Theorem 1 is established. The proof is obtained. □

Remark 1. 

When $B=+\infty$, it can be concluded from Theorem 1 that in terms of each $\lambda \in (0,\frac{1}{A^{*}})$, we obtain the same result as Theorem 1.

Put

$$A^{*}=\underset{t\to +\infty }{lim\; inf}\frac{{\displaystyle \sum _{\nu =1}^{\tau }}{\displaystyle \sum _{\kappa =1}^{\gamma }}\underset{\left|\epsilon \right|\le t}{max}F((\kappa ,\nu ),\epsilon )}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}.$$

If $A^{*}=0$, we regard $\frac{1}{A^{*}}$ as $+\infty$.

Corollary 1. 

Assume

$$A^{*}<\frac{pB}{2b{(\gamma +\tau )}^2},$$

(12)

for each $\lambda \in (\frac{2b{(\gamma +\tau )}^2}{pB},\frac{1}{A^{*}})$, we obtain the same result as Theorem 1.

Remark 2. 

With $B=+\infty$, it can be concluded from Corollary 1 that concerning each $\lambda \in (0,\frac{1}{A^{*}})$, we obtain the same result as Theorem 1. With $A^{*}=0$, in accordance with Corollary 1, we consider that in terms of each $\lambda \in \left(\frac{2b{(\gamma +\tau )}^2}{pB},+\infty \right)$, we obtain the same result as Theorem 1. With $B=+\infty$ and $A^{*}=0$, we consider that in terms of each $\lambda >0$, we obtain the same result as Theorem 1.

When $f\left(\right(\kappa ,\nu ),u(\kappa ,\nu \left)\right)=\alpha (\kappa ,\nu )h\left(u\right(\kappa ,\nu \left)\right)$, we regard problem DKP as DKP ${}^{*}$, namely

$$\begin{array}{c}{-(a+b\parallel u\parallel }^p)({\mathrm{\Delta }}_1\left({\varphi }_p\left({\mathrm{\Delta }}_{1}u(\kappa -1,\nu )\right)\right)+{\mathrm{\Delta }}_2\left({\varphi }_p\left({\mathrm{\Delta }}_{2}u(\kappa ,\nu -1)\right)\right))=\lambda \alpha (\kappa ,\nu )h\left(u(\kappa ,\nu )\right),(\kappa ,\nu )\in [1,\gamma ]\times [1,\tau ],\hfill \end{array}$$

with boundary conditions (1), where $\alpha :\mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau )\to \mathbb{R}$ represents a non-negative and non-zero function and $h:[0,+\infty )\to \mathbb{R}$ indicates a non-negative continuous function, thus enabling $h\left(0\right)=0$.

Corollary 2. 

Assume

$$\underset{t\to +\infty }{lim\; inf}\frac{{\int }_0^{t}h\left(s\right)ds}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}<\frac{p}{2b{(\gamma +\tau )}^2}\underset{t\to +\infty }{lim\; sup}\left(\frac{{\int }_0^{t}h\left(s\right)ds}{t^{2p}}\right).$$

Then for every

$$\lambda \in \frac{1}{{\sum }_{\nu =1}^{\tau }{\sum }_{\kappa =1}^{\gamma }\alpha (\kappa ,\nu )}\left[\frac{2b{(\gamma +\tau )}^2}{p{lim\; sup}_{t\to +\infty }\left(\frac{{\int }_0^{t}h\left(s\right)ds}{t^{2p}}\right)},\frac{1}{{lim\; inf}_{t\to +\infty }\frac{{\int }_0^{t}h\left(s\right)ds}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}}\right],$$

problem DKP ${}^{*}$ admits a positive solution sequence that is unbounded.

Proof. 

Let

$$f((\kappa ,\nu ),u)=\left\{\begin{array}{cc}\alpha (\kappa ,\nu )h\left(u\right),\hfill & u\ge 0,\hfill \\ 0,\hfill & u<0,\hfill \end{array}\right.$$

for every $(\kappa ,\nu )\in \mathbb{Z}(1,\gamma )\times \mathbb{Z}(1,\tau ),u\in \mathbb{R}$, from Corollary 1 and Lemma 2, we conclude Corollary 2 holds. □

4. An Example

Example 1. 

Let $\gamma =\tau =2,\epsilon =\frac{1}{100},a=b=1$. Set $\alpha (\kappa ,\nu )=1$, for $(\kappa ,\nu )\in \mathbb{Z}(1,2)\times \mathbb{Z}(1,2)$ and let $h:[0,+\infty )\to \mathbb{R}$,

$$h\left(t\right)=\left\{\begin{array}{cc}2p{t}^{2p-1}(2p+\epsilon +2pcos(\epsilon lnt)-\epsilon sin(\epsilon lnt)),\hfill & t>0,\hfill \\ 0,\hfill & t=0.\hfill \end{array}\right.$$

Then

$$\begin{array}{cc}\hfill \underset{t\to +\infty }{lim\; inf}\frac{{\int }_0^{t}h\left(s\right)ds}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}& =\underset{t\to +\infty }{lim\; inf}\frac{t^{2p}(2p+\epsilon +2pcos(\epsilon lnt))}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}\hfill \\ \hfill & =\underset{t\to +\infty }{lim\; inf}\frac{t^p(2p+\epsilon +2pcos(\epsilon lnt))}{\frac{b{4}^{2p}}{2p{(\gamma +\tau +2)}^{2p-2}}{t}^p}\hfill \\ \hfill & =\frac{\epsilon 2p{(\gamma +\tau +2)}^{2p-2}}{4^{2p}b}\approx 0.005625\hfill \end{array}$$

and

$$\underset{t\to +\infty }{lim\; sup}\left(\frac{{\int }_0^{t}h\left(s\right)ds}{t^{2p}}\right)=\underset{t\to +\infty }{lim\; sup}\frac{t^{2p}(2p+\epsilon +2pcos(\epsilon lnt))}{t^{2p}}=8.01.$$

Due to

$$\frac{2}{2\times {(2+2)}^2}>\frac{0.005625}{8.01},$$

one has

$$\underset{t\to +\infty }{lim\; inf}\frac{{\int }_0^{t}h\left(s\right)ds}{\frac{2a{4}^p{t}^p{(r+\tau +2)}^{p-1}+b{4}^{2p}{t}^{2p}}{2p{(r+\tau +2)}^{2p-2}}}<\frac{2}{2\times {(2+2)}^2}\underset{t\to +\infty }{lim\; sup}\left(\frac{{\int }_0^{t}h\left(s\right)ds}{t^{2p}}\right).$$

According to Corollary 2, for every $\lambda \in (0.4994,44.4445)$, the problem we consider leads to the same conclusion as Corollary 2.

5. Conclusions

In this paper, we attempt to study the existence of infinitely many solutions for the partial discrete Kirchhoff-type problems involving p-Laplacian for the first time. To prove the existence of infinitely many solutions of the partial discrete Kirchhoff-type problems involving p-Laplacian, we use the critical point theory. Further, by virtue of the strong maximum principle we have established, we acquire some sufficient conditions for the presence of infinitely many positive solutions to the boundary value problems. The discrete Kirchhoff-type problem involving p-Laplacian has strong theoretical significance and application value. This work solves the existence of infinitely many solutions to the boundary value problem of the Kirchhoff type partial difference equation, and three positive solutions of the Kirchhoff type partial difference equation can be studied as future research problems.