Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

Abstract

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving $\mathit{p}$-mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

1. Introduction

Let $\mathbb{Z},\mathbb{N}$ and $\mathbb{R}$ denote the sets of integer numbers, natural numbers and real numbers, respectively. For $a,b\in \mathbb{Z}$, define $\mathbb{Z}(a)=\{a,a+1,\cdots \}$, and $\mathbb{Z}(a,b)=\{a,a+1,\cdots ,b\}$ when $a\le b$.

Consider the following Dirichlet boundary value problem of the nonlinear difference equation

$$(D_p^{\lambda ,f})\left\{\begin{array}{c}-▵\left({\varphi }_{p,c}\left(▵u(k-1)\right)\right)=\lambda f(k,u(k)),k\in \mathbb{Z}(1,T),\hfill \\ u(0)=u(T+1)=0,\hfill \end{array}\right.$$

where T is a given positive integer, $\lambda$ is a positive real parameter, ▵ is the forward difference operator defined by $▵u(k)=u(k+1)-u(k)$, $f(k,·):\mathbb{R}\to \mathbb{R}$ is a continuous function for each $k\in \mathbb{Z}(1,T)$ and ${\varphi }_{p,c}(s):={(1+|s|}^2{)}^{\frac{p-2}{2}}s,p\in [1,+\infty )$. Here, $▵\left({\varphi }_{p,c}\left(▵u(k-1)\right)\right)$ may be seen as a discretization of the p-mean curvature operator.

We may think problem $(D_p^{\lambda ,f})$ as being a discrete analog of one-dimensional case of the following problem

$$\left\{\begin{array}{cc}-\mathrm{div}\left({\varphi }_{p,c}\left(▿u\right)\right)=\lambda f(x,u),\hfill & x\in \Omega \subset {\mathbb{R}}^n,\hfill \\ u=0,\hfill & x\in \partial \Omega ,\hfill \end{array}\right.$$

(1)

where $div\left({\varphi }_{p,c}\left(▿u\right)\right)$ is named $\mathit{p}$-mean curvature operator, which is a generalization of mean curvature operator; see [1,2]. If $p=1$, it reduces to the mean curvature operator. If $p=2$, it reduces to the Laplacian operator. The above problem arises from differential geometry and physics such as capillarity; see [3,4,5] and references therein. When $p=1$ and $f(x,u)=u$, the above problem describes the free surface of a pendent drop filled with liquid under gravitational field [4]. In the past decades, several authors have discussed the existence and multiplicity of solutions of Problem (1); see [1,6,7,8,9,10,11,12]. For example, Chen and Shen in [1] have obtained the existence of infinitely many solutions of Problem (1) with $\lambda =1$ via a symmetric version of Mountain Pass Theorem. When $p=1$ and $\Omega =(0,1)$, Obersnel and Omari in [11] have established the existence and multiplicity of positive solutions of Problem (1), which depend on the behavior of f at zero or at infinity. G. A. Afrouzi et al. in [6] have acquired a sequence of nonnegative and nontrivial solutions strongly converging to zero in $C^1([0,1])$, under suitable oscillating behavior of the nonlinear term f at zero. However, the results on the existence of solutions for problem $(D_p^{\lambda ,f})$ are scarce in the literature besides the case of $p=1$.

Nonlinear discrete problems appear in many mathematical models, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics, fluid mechanics and many others; see [13,14,15,16,17]. Many authors have discussed the existence and multiplicity of solutions for difference equations through classical tools of nonlinear analysis: Fixed point theorems, upper and lower solutions techniques; see [7,9] and the references given therein. Since 2003, by starting from the seminal paper [18], variational methods have been used to investigate nonlinear difference equations, which have obtained various results; see [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34].

In paper [35], the authors have considered problem $(D_1^{\lambda ,f})$, obtaining infinitely many positive solutions when $\lambda$ belongs to a precise real interval. It is worth noticing that the suitable oscillating behaviors of the nonlinear term f at infinity play a key role. Inspired by [19,32,35,36,37,38,39,40], the main purpose of this paper is to investigate the existence conditions of infinitely many constant-sign solutions for problem $(D_p^{\lambda ,f})$, without any symmetry hypothesis. Here, a solution $\{u(k)\}$ of $(D_p^{\lambda ,f})$ is called a constant-sign solution, if $u(k)>0$ for all $k\in \mathbb{Z}(1,T)$ or $u(k)<0$ for all $k\in \mathbb{Z}(1,T)$. Compared to problem $(D_1^{\lambda ,f})$, problem $(D_p^{\lambda ,f})$ is more difficult to handle. To facilitate the analysis, we have to divide the problem into two categories: $1\le p<2$ and $2\le p<+\infty$. We believe that this is the first time to discuss the existence of infinitely many solutions for a non-linear second order difference equation with $p$-mean curvature operator.

A special case of our results is the following.

Theorem 1.

Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $g(t)t\ge 0$ for $t\ne 0$. Assume that

$$\underset{t\to \infty }{lim\; inf}\frac{{\int }_0^{t}g(\tau )d\tau }{{\left|t\right|}^p}=0,\mathrm{and}\underset{t\to \infty }{lim\; sup}\frac{{\int }_0^{t}g(\tau )d\tau }{{\left|t\right|}^p}=+\infty .$$

Then, for every $\lambda >0$, the problem

$$\left\{\begin{array}{c}-▵\left({\varphi }_{p,c}\left(▵u(k-1)\right)\right)=\lambda g(u(k)),k\in \mathbb{Z}(1,T),\hfill \\ u(0)=u(T+1)=0,\hfill \end{array}\right.$$

admits two unbounded sequences of constant-sign solutions (one positive and one negative).

This paper is organized as follows. In Section 2, we introduce the the suitable Banach space and appropriate functional corresponding to problem $(D_p^{\lambda ,f})$. To obtain sequences of constant-sign solutions of problem $(D_p^{\lambda ,f})$, three basic lemmas are introduced. In Section 3, under suitable hypotheses on f, we obtain the existence of infinitely many constant-sign solutions for problem $(D_p^{\lambda ,f})$. In Section 4, we give two examples to demonstrate our results. Finally, conclusions are given for this paper.

2. Mathematical Background

To solve problem $(D_p^{\lambda ,f})$, we naturally select the T-dimensional Banach space

$$X=\{u:\mathbb{Z}(0,T+1)\to \mathbb{R}:u(0)=u(T+1)=0\},$$

endowed with the norm

$$\parallel u\parallel :={\left({\displaystyle \sum _{k=1}^T}{(▵u(k))}^2\right)}^{\frac{1}{2}}forallu\in X.$$

Another useful norm on X is

$${\left|\right|u\left|\right|}_{\infty }:=\underset{k\in \mathbb{Z}(1,T)}{max}\left|u(k)\right|\mathrm{for}\mathrm{all}u\in X.$$

In the sequel, we will use the following inequalities.

For $0<r<s,x_k\ge 0,k\in \mathbb{Z}(1,n)$, one has

$${\left({\displaystyle \sum _{k=1}^n}{x}_k^s\right)}^{1/s}\le {\left({\displaystyle \sum _{k=1}^n}{x}_k^r\right)}^{1/r},$$

(2)

see [41].

$${\left|\right|u\left|\right|}_{\infty }\le \frac{\sqrt{T+1}}{2}\parallel u\parallel ,$$

(3)

for every $u\in X$, it can follow from Lemma 2.2 of [42].

For all $u\in X$, let

$$\mathsf{\Phi }(u):=\frac{1}{p}{\displaystyle \sum _{k=0}^T}\left({\left(1+{(▵u(k))}^2\right)}^{\frac{p}{2}}-1\right),and\mathsf{\Psi }(u):={\displaystyle \sum _{k=1}^T}F(k,u(k)),$$

(4)

where $F(k,t):={\int }_0^{t}f(k,\tau )d\tau$ for every $t\in \mathbb{R}$ and $k\in \mathbb{Z}(1,T)$. Further, let us denote $I_{\lambda }(u):=\mathsf{\Phi }(u)-\lambda \mathsf{\Psi }(u)$ for $u\in X.$ Through standard arguments, we follow that $I_{\lambda }\in C^1(S,\mathbb{R})$, and the critical points of $I_{\lambda }$ are exactly the solutions of problem $(D_p^{\lambda ,f})$. In fact, one has

$$\begin{array}{ll}\hfill I_{\lambda }^{\prime }(u)(v)& ={\displaystyle \sum _{k=0}^T}\left({\varphi }_{p,c}(▵u(k)\right)▵v(k)-\lambda {\displaystyle \sum _{k=1}^T}f(k,u(k))v(k)\hfill \\ & ={\displaystyle \sum _{k=0}^T}\left({\varphi }_{p,c}(▵u(k)\right)v(k+1)-{\displaystyle \sum _{k=0}^T}\left({\varphi }_{p,c}(▵u(k)\right)v(k)-\lambda {\displaystyle \sum _{k=1}^T}f(k,u(k))v(k)\hfill \\ & ={\displaystyle \sum _{k=1}^T}\left({\varphi }_{p,c}(▵u(k-1)\right)v\left(k\right)-{\displaystyle \sum _{k=1}^T}\left({\varphi }_{p,c}(▵u\left(k\right)\right)v\left(k\right)-\lambda {\displaystyle \sum _{k=1}^T}f(k,u\left(k\right))v\left(k\right)\hfill \\ & =-{\displaystyle \sum _{k=1}^T}[▵(\left({\varphi }_{p,c}(▵u\left(k\right)\right)-\lambda f(k,u\left(k\right))]v\left(k\right),\hfill \end{array}$$

for all $u,v\in X.$

Next, we need to establish the following strong maximum principle to obtain the positive solutions of problem $\left(D_p^{\lambda ,f}\right)$, i.e., $u\left(k\right)>0$ for each $k\in \mathbb{Z}(1,T)$.

Lemma 1.

Assume $u\in X$ such that either

$$u\left(k\right)>0or-▵\left({\phi }_{p,c}(▵u(k-1)\right)\ge 0,$$

(5)

for any $k\in \mathbb{Z}(1,T)$. Then, either $u>0$ in $\mathbb{Z}(1,T)$ or $u\equiv 0$.

Proof. For $u\in X$, put $m=min\{u(k),k\in \mathbb{Z}(0,T+1)\}$, then $m\le 0$.

If there exists $j\in \mathbb{Z}(1,T)$ such that $u(j)=m$, we claim that $u\equiv 0$. Indeed, since $\Delta u(j-1)=u(j)-u(j-1)\le 0$ and $\Delta u(j)=u(j+1)-u(j)\ge 0$, ${\phi }_{p,c}(s)$ is strictly monotone increasing in s, and ${\phi }_{p,c}(0)=0$, we have

$${\phi }_{p,c}(▵u(j))\ge 0\ge {\phi }_{p,c}(▵u(j-1)).$$

(6)

On the other hand, by (5), let $k=j$, we obtain

$${\phi }_{p,c}(▵u(j))\le {\phi }_{p,c}(▵u(j-1)).$$

(7)

Combining inequalities (6) and (7), we get that ${\phi }_{p,c}(▵u(j))=0={\phi }_{p,c}(▵u(j-1))$. That is $u(j+1)=u(j-1)=u(j)=m$. By iterating this argument, we obtain easily $u(0)=u(1)=u(2)=\dots =u(T)=u(T+1)$. Thus $u\equiv 0$.

If $u(j)>m$ for every $j\in \mathbb{Z}(1,T)$, then $u(0)=u(T+1)=m=0$. It follows that $u(j)>0$, for all $j\in \mathbb{Z}(1,T)$. The proof is complete.

In the same way, we have the following result to get negative solutions problem $(D_p^{\lambda ,f})$, i.e., $u(k)<0$ for each $k\in \mathbb{Z}(1,T)$.

Lemma 2.

Assume $u\in X$ such that either

$$u(k)<0or-▵\left({\phi }_{p,c}(▵u(k-1)\right)\le 0,$$

(8)

for any $k\in \mathbb{Z}(1,T)$. Then, either $u<0$ in $\mathbb{Z}(1,T)$ or $u\equiv 0$.

Truncation techniques are usually used to discuss the existence of constant-sign solutions. To the end, we introduce the following truncations of the functions $f(k,t)$ for every $k\in \mathbb{Z}(1,T).$

If $f(k,0)\ge 0$ for each $k\in \mathbb{Z}(1,T)$. Set

$$f^{+}(k,t):=\left\{\begin{array}{ccc}f(k,t),& & \mathrm{if}t\ge 0,\\ f(k,0),& & \mathrm{if}t<0.\end{array}\right.$$

Clearly, $f^{+}(k,·)$ is also continuous, for every $k\in \mathbb{Z}(1,T)$. By Lemma 1, all solutions of problem $(D_p^{\lambda ,f^{+}})$ are also solutions of problem $(D_p^{\lambda ,f})$. Therefore, when problem $(D_p^{\lambda ,f^{+}})$ has non-zero solutions, then problem $(D_p^{\lambda ,f})$ possesses positive solutions.

If $f(k,0)\le 0$ for each $k\in \mathbb{Z}(1,T)$. Set

$$f^{-}(k,t):=\left\{\begin{array}{ccc}f(k,0),& & \mathrm{if}t>0,\\ f(k,t),& & \mathrm{if}t\le 0.\end{array}\right.$$

When problem $(D_p^{\lambda ,f^{-}})$ has non-zero solutions, then problem $(D_p^{\lambda ,f})$ possesses negative solutions.

Here, we introduce a lemma (Theorem 4.3 of [38]) which is the main tool used to research problem $(D_p^{\lambda ,f}).$

Lemma 3.

Let X be a finite dimensional Banach space and let $I_{\lambda }:X\to \mathbb{R}$ be a function satisfying the following structure hypothesis:

(H)$I_{\lambda }(u):=\mathsf{\Phi }(u)-\lambda \mathsf{\Psi }(u)$ for all $u\in X$, where $\mathsf{\Phi },\mathsf{\Psi }:X\to \mathbb{R}$ be two continuously G$\widehat{a}$teux differentiable functions with Φ coercive, i.e., ${lim}_{\parallel u\parallel \to +\infty }\mathsf{\Phi }(u)=+\infty$, and such that ${inf}_X\mathsf{\Phi }=\mathsf{\Phi }(0)=\mathsf{\Psi }(0)=0.$

For all $r>0$, put

$$\phi (r):=\frac{\underset{{\mathsf{\Phi }}^{-1}[0,r]}{sup}\mathsf{\Psi }}{r},and{\phi }_{\infty }:=\underset{r\to +\infty }{lim\; inf}\phi (r).$$

Assume that ${\phi }_{\infty }<+\infty$ and for each $\lambda \in (0,\frac{1}{{\phi }_{\infty }})$$I_{\lambda }$ is unbounded from below. Then, there is a sequence $\left\{u_n\right\}$ of critical points $(localminima)$ of $I_{\lambda }$ such that $\underset{n\to +\infty }{lim}\mathsf{\Phi }(u_n)=+\infty$.

3. Main Results

In the following, we will discuss the existence of constant-sign solutions of problem $(D_p^{\lambda ,f})$. Our purpose is to apply Lemma 3 to the function $I_{\lambda }^{\pm }:X\to \mathbb{R},$$I_{\lambda }^{\pm }(u):=\mathsf{\Phi }(u)-\lambda {\mathsf{\Psi }}^{\pm }(u),$ where ${\mathsf{\Psi }}^{\pm }(u)={\sum }_{k=1}^T{F}^{\pm }(k,u(k))$ and $F^{\pm }(k,t):={\int }_0^t{f}^{\pm }(k,\tau )d\tau$ for every $k\in \mathbb{Z}(1,T)$ and then exploit Lemma 1 or Lemma 2 to get our results.

Let

$$A_{\pm \infty }:=\underset{t\to +\infty }{lim\; inf}\frac{{\displaystyle {\displaystyle \sum _{k=1}^T}\underset{0\le s\le t}{max}F(k,\pm s)}}{t^p},\mathrm{and}{B}^{\pm \infty }:=\underset{t\to \pm \infty }{lim\; sup}\frac{{\displaystyle {\displaystyle \sum _{k=1}^T}F(k,t)}}{{\left|t\right|}^p}.$$

Considering the functional $I_{\lambda }^{+}$, we have the following conclusions.

Theorem 2.

Let $1\le p<2$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)\ge 0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_1)A_{+\infty }<\frac{2^{p-1}}{{(T+1)}^{\frac{p}{2}}}{B}^{+\infty }.}$$

Then, for each $\lambda \in (\frac{2}{p{B}^{+\infty }},\frac{2^p}{p{(T+1)}^{\frac{p}{2}}{A}_{+\infty }})$, problem $(D_p^{\lambda ,f})$ has an unbounded sequence of positive solutions.

Proof. Consider the auxiliary problem

$$(D_p^{\lambda ,f^{+}})\left\{\begin{array}{c}-▵\left({\varphi }_{p,c}\left(▵u(k-1)\right)\right)=\lambda f^{+}(k,u(k)),k\in \mathbb{Z}(1,T),\hfill \\ u(0)=u(T+1)=0.\hfill \end{array}\right.$$

Obviously $\mathsf{\Phi }$ and ${\mathsf{\Psi }}^{+}$ satisfy hypothesis required in Lemma 3. For $t>0$, set

$$r=\frac{1}{p}{\left(\sqrt{\frac{4t^2}{T+1}+{(T+1)}^{\frac{2p-2}{p}}}-{(T+1)}^{\frac{p-1}{p}}\right)}^p.$$

Assume $u\in X$ and

$$\mathsf{\Phi }(u)=\frac{1}{p}{\displaystyle \sum _{k=0}^T}\left({\left(1+{(▵u(k))}^2\right)}^{\frac{p}{2}}-1\right)\le r.$$

Put $v(k)={\left(1+{(▵u(k))}^2\right)}^{\frac{p}{2}}-1$, for every $k\in \mathbb{Z}(0,T)$, then ${\displaystyle \sum _{k=0}^T}v(k)\le pr$.

By (2) and Hölder inequality as well, we have

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{k=0}^T}{(▵u(k))}^2& =& {\displaystyle \sum _{k=0}^T}\left({\left({(1+v(k))}^{\frac{1}{p}}\right)}^2-1\right)\hfill \\ & \le & {\left({\displaystyle \sum _{k=0}^T}v(k)\right)}^{\frac{2}{p}}+2{(T+1)}^{\frac{p-1}{p}}{\left({\displaystyle \sum _{k=0}^T}v(k)\right)}^{\frac{1}{p}}\hfill \\ & \le & {(pr)}^{\frac{2}{p}}+2{(T+1)}^{\frac{p-1}{p}}{(pr)}^{\frac{1}{p}}\hfill \\ & =& \frac{4t^2}{T+1}.\hfill \end{array}$$

Owing to (3), it follows

$${\left|\right|u\left|\right|}_{\infty }\le \frac{\sqrt{T+1}}{2}{\left({\displaystyle \sum _{k=0}^T}{(▵u(k))}^2\right)}^{\frac{1}{2}}\le t.$$

Thus, one has ${\mathsf{\Phi }}^{-1}[0,r]\subseteq {\{u\in X:|\left|u\right||}_{\infty }\le t\}$

By the definition of $\phi$, we obtain

$$\phi (r)=\frac{\underset{{\mathsf{\Phi }}^{-1}[0,r]}{sup}{\mathsf{\Psi }}^{+}}{r}\le \frac{{\displaystyle \underset{{\left|\right|u\left|\right|}_{\infty }\le t}{sup}{\displaystyle \sum _{k=0}^T}{F}^{+}(k,u(k))}}{r}\le \frac{{\displaystyle p{\displaystyle \sum _{k=1}^T}\underset{0\le s\le t}{max}F(k,s)}}{{\left(\sqrt{\frac{4t^2}{T+1}+{(T+1)}^{\frac{2p-2}{p}}}-{(T+1)}^{\frac{p-1}{p}}\right)}^p}.$$

Bearing in mind condition$(i_1)$, we follow that ${\phi }_{\infty }\le \frac{p{(T+1)}^{\frac{p}{2}}}{2^p}{A}_{+\infty }<+\infty .$

In the next step, we need to prove that $I_{\lambda }^{+}$ is unbounded from below. To this end, we consider two cases: $B^{+\infty }=+\infty$ and $B^{+\infty }<+\infty$. If $B^{+\infty }=+\infty$, let $\left\{c_n\right\}$ be a sequence of positive numbers, with ${lim}_{n\to +\infty }{c}_n=+\infty$, such that

$${\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)={\displaystyle \sum _{k=1}^T}F(k,c_n)\ge \frac{(2+p)}{\lambda p}{c}_n^p,foreveryn\in \mathbb{N}.$$

In the following, we take in X the sequence $\left\{{\omega }_n\right\}$ defined by putting ${\omega }_n(k)=c_n$, for $k\in \mathbb{Z}(1,T)$.

Using again (2), one has

$$I_{\lambda }^{+}({\omega }_n)=\frac{2}{p}\left({\left(1+c_n^2\right)}^{\frac{p}{2}}-1\right)-\lambda {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)\le \frac{2}{p}{c}_n^p-\frac{2+p}{p}{c}_n^p=-c_n^p,$$

which implies that ${lim}_{n\to +\infty }{I}_{\lambda }^{+}({\omega }_n)=-\infty$. If $B^{+\infty }<+\infty$, since $\lambda >\frac{2}{p{B}^{+\infty }}$, we may take ${ϵ}_0>0$ such that $\frac{2}{p}-\lambda B^{+\infty }+\lambda {ϵ}_0<0$. Then there exists a sequence of positive numbers $\left\{c_n\right\}$ such that ${lim}_{n\to +\infty }{c}_n=+\infty$ and

$$(B^{+\infty }-{ϵ}_0)c_n^p\le {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)={\displaystyle \sum _{k=1}^T}F(k,c_n)\le (B^{+\infty }+{ϵ}_0)c_n^p.$$

Arguing as before and by choosing $\left\{{\omega }_n\right\}$ in X as above, we have

$$I_{\lambda }^{+}({\omega }_n)=\frac{2}{p}\left({\left(1+c_n^2\right)}^{\frac{p}{2}}-1\right)-\lambda {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)\le \frac{2}{p}{c}_n^p-\lambda (B^{+\infty }-{ϵ}_0)c_n^p=(\frac{2}{p}-\lambda B^{+\infty }+\lambda {ϵ}_0)c_n^p.$$

Since $\frac{2}{p}-\lambda B^{+\infty }+\lambda {ϵ}_0<0$, it is clear that ${lim}_{n\to +\infty }{I}_{\lambda }^{+}({\omega }_n)=-\infty$. Considering the above two cases, we follow that $I_{\lambda }^{+}$ is unbounded from below.

According to Lemma 3, there exist a sequence $\left\{u_n\right\}$ of critical points (local minima) of $I_{\lambda }^{+}$ such that $\underset{n\to +\infty }{lim}\mathsf{\Phi }(u_n)=+\infty$. Hence, for every $n\in \mathbb{N}$, $u_n$ is a non-zero solution of problem $(D_p^{\lambda ,f^{+}})$, by Lemma 1, $u_n$ is a positive solution of problem $(D_p^{\lambda ,f})$. Since $\mathsf{\Phi }$ is bounded on bounded sets and $\underset{n\to +\infty }{lim}\mathsf{\Phi }(u_n)=+\infty$, $\left\{u_n\right\}$ must be unbounded. So Theorem 2 holds and the proof is complete.

Theorem 3.

Let $2\le p<+\infty$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)\ge 0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_2)A_{+\infty }<\frac{{(\sqrt{2})}^p}{{(T+1)}^{p-1}}{B}^{+\infty }.}$$

Then, for each $\lambda \in (\frac{{(\sqrt{2})}^p}{p{B}^{+\infty }},\frac{2^p}{p{(T+1)}^{p-1}{A}_{+\infty }})$, problem $(D_p^{\lambda ,f})$ has an unbounded sequence of positive solutions.

Proof. We sketch only the differences with the proof of Theorem 2. For $t>0$, make

$$r=\frac{{(2t)}^p}{p{(T+1)}^{p-1}}.$$

Assume $u\in X$ and

$$\mathsf{\Phi }(u)=\frac{1}{p}{\displaystyle \sum _{k=0}^T}\left({\left(1+{(▵u(k))}^2\right)}^{\frac{p}{2}}-1\right)\le r.$$

Denote $v(k)={\left(1+{(▵u(k))}^2\right)}^{\frac{p}{2}}-1$, for every $k\in \mathbb{Z}(0,T)$, then ${\displaystyle \sum _{k=0}^T}v(k)\le pr$.

Noting the inequality ${(x+y)}^{\theta }\le x^{\theta }+y^{\theta }$, for $0<\theta \le 1,x\ge 0,y\ge 0$ and Hölder inequality, one has

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{k=0}^T}{(▵u(k))}^2& =& {\displaystyle \sum _{k=0}^T}{\left(1+v(k)\right)}^{\frac{2}{p}}-1)\hfill \\ & \le & {\displaystyle \sum _{k=0}^T}{(v(k))}^{\frac{2}{p}}\hfill \\ & \le & {(T+1)}^{\frac{p-2}{p}}{\left({\displaystyle \sum _{k=0}^T}v(k)\right)}^{\frac{2}{p}}\hfill \\ & \le & {(T+1)}^{\frac{p-2}{p}}{(pr)}^{\frac{2}{p}}=\frac{4t^2}{T+1}.\hfill \end{array}$$

Applying (3), we have

$${\left|\right|u\left|\right|}_{\infty }\le \frac{\sqrt{T+1}}{2}{\left({\displaystyle \sum _{k=0}^T}{(▵u_k)}^2\right)}^{\frac{1}{2}}\le t.$$

By the definition of $\phi$, we have

$$\phi (r)=\frac{\underset{{\mathsf{\Phi }}^{-1}[0,r]}{sup}{\mathsf{\Psi }}^{+}}{r}\le \frac{{\displaystyle \underset{{\left|\right|u\left|\right|}_{\infty }\le t}{sup}{\displaystyle \sum _{k=0}^T}{F}^{+}(k,u(k))}}{r}\le \frac{{\displaystyle p{(T+1)}^{p-1}{\displaystyle \sum _{k=1}^T}\underset{0\le s\le t}{max}F(k,s)}}{2^p{t}^p}.$$

Using condition$(i_2)$, ${\phi }_{\infty }\le \frac{p{(T+1)}^{p-1}}{2^p}{A}_{+\infty }<+\infty$ holds.

Now, we verify that $I_{\lambda }^{+}$ is unbounded form blow. Fist, assume that $B^{+\infty }=+\infty$. Let $\left\{c_n\right\}$ be a sequence of positive numbers, with ${lim}_{n\to +\infty }{c}_n=+\infty$, such that

$${\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)={\displaystyle \sum _{k=1}^T}F(k,c_n)\ge \frac{{(\sqrt{2})}^p+p}{\lambda p}{c}_n^p,forn\in \mathbb{N}.$$

Picking the sequence $\left\{{\omega }_n\right\}$ in X by ${\omega }_n(k)=c_n,$ for $k\in \mathbb{Z}(1,T).$ Exploiting the inequality ${(x+y)}^{\theta }\le 2^{\theta -1}(x^{\theta }+y^{\theta })$ for $\theta \ge 1,x\ge 0,y\ge 0$ , we get

$$\begin{array}{ccc}{I}_{\lambda }^{+}({\omega }_n)\hfill & =\hfill & \frac{2}{p}\left({\left(1+c_n^2\right)}^{\frac{p}{2}}-1\right)-\lambda {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)\le \frac{{(\sqrt{2})}^p}{p}{c}_n^p+\frac{{(\sqrt{2})}^p-2}{p}-\frac{{(\sqrt{2})}^p+p}{p}{c}_n^p\hfill \\ & =\hfill & -c_n^p+\frac{{(\sqrt{2})}^p-2}{p},\hfill \end{array}$$

which implies that ${lim}_{n\to +\infty }{I}_{\lambda }({\omega }_n)=-\infty$.

Next, assume that $B^{+\infty }<+\infty$. Since $\lambda >\frac{{(\sqrt{2})}^p}{p{B}^{+\infty }}$, we may take ${ϵ}_0>0$ such that $\frac{{(\sqrt{2})}^p}{p}-\lambda B^{\infty }+\lambda {ϵ}_0<0$. Then there exists a sequence of positive numbers $\left\{c_n\right\}$ such that ${lim}_{n\to +\infty }{c}_n=+\infty$ and

$$(B^{+\infty }-{ϵ}_0)c_n^p\le {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)={\displaystyle \sum _{k=1}^T}F(k,c_n)\le (B^{+\infty }+{ϵ}_0)c_n^p.$$

Define the sequence $\left\{{\omega }_n\right\}$ in S as above, we obtain

$$\begin{array}{ccc}{I}_{\lambda }^{+}({\omega }_n)\hfill & =\hfill & \frac{2}{p}\left({\left(1+c_n^2\right)}^{\frac{p}{2}}-1\right)-\lambda {\displaystyle \sum _{k=1}^T}{F}^{+}(k,c_n)\le \frac{{(\sqrt{2})}^p}{p}{c}_n^p+\frac{{(\sqrt{2})}^p-2}{p}-\lambda (B^{+\infty }-{ϵ}_0)c_n^p\hfill \\ & =\hfill & (\frac{{(\sqrt{2})}^p}{p}-\lambda B^{+\infty }+\lambda {ϵ}_0)c_n^p+\frac{{(\sqrt{2})}^p-2}{p}.\hfill \end{array}$$

Since $\frac{{(\sqrt{2})}^p}{p}-\lambda B^{+\infty }+\lambda {ϵ}_0<0$, it is obvious that ${lim}_{n\to +\infty }{I}_{\lambda }^{+}({\omega }_n)=-\infty$.

Thus, we follow that $I_{\lambda }^{+}$ is unbounded from below. According to Lemmas 1 and 3, we have finished the proof of the theorem.

Similarly, considering the functional $I_{\lambda }^{-}$, we can achieve the following results.

Theorem 4.

Let $1\le p<2$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)\le 0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_3)A_{-\infty }<\frac{2^{p-1}}{{(T+1)}^{\frac{p}{2}}}{B}^{-\infty }.}$$

Then, for each $\lambda \in (\frac{2}{p{B}^{-\infty }},\frac{2^p}{p{(T+1)}^{\frac{p}{2}}{A}_{-\infty }})$, problem $(D_p^{\lambda ,f})$ has an unbounded sequence of negative solutions.

Theorem 5.

Let $2\le p<+\infty$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)\le 0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_4)A_{-\infty }<\frac{{(\sqrt{2})}^p}{{(T+1)}^{p-1}}{B}^{-\infty }.}$$

Then, for each $\lambda \in (\frac{{(\sqrt{2})}^p}{p{B}^{-\infty }},\frac{2^p}{p{(T+1)}^{p-1}{A}_{-\infty }})$, problem $(D_p^{\lambda ,f})$ has an unbounded sequence of negative solutions.

Combining Theorems 2 and 4, we have the following corollary.

Corollary 1.

Let $1\le p<2$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)=0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_5)max\{A_{+\infty },A_{-\infty }\}<\frac{2^{p-1}}{{(T+1)}^{\frac{p}{2}}}min\{B^{+\infty },B^{-\infty }\}.}$$

Then, for each $\lambda \in (\frac{2}{pmin\{B^{+\infty },B^{-\infty }\}},\frac{2^p}{p{(T+1)}^{\frac{p}{2}}max\{A_{+\infty },A_{-\infty }\}})$, problem $(D_p^{\lambda ,f})$ admits two unbounded sequences of constant-sign solutions ( one positive and one negative ).

Similarly, combining Theorems 3 and 5, we have the following corollary.

Corollary 2.

Let $2\le p<+\infty$ and $f(k,·):\mathbb{R}\to \mathbb{R}$ to be a continuous function with $f(k,0)=0$ for each $k\in \mathbb{Z}(1,T)$. Assume that

$${\displaystyle (i_6)max\{A_{+\infty },A_{-\infty }\}<\frac{{(\sqrt{2})}^p}{{(T+1)}^{p-1}}min\{B^{+\infty },B^{-\infty }\}.}$$

Then, for each $\lambda \in (\frac{{(\sqrt{2})}^p}{pmin\{B^{+\infty },B^{-\infty }\}},\frac{2^p}{p{(T+1)}^{p-1}max\{A_{+\infty },A_{-\infty }\}})$, problem $(D_p^{\lambda ,f})$ admits admits two unbounded sequences of constant-sign solutions ( one positive and one negative ).

Remark 1.

If we let $p\to 2^{-}$ in Theorem 2, we find that the conditions and consequence of Theorem 2 is the same as those of Theorem 3 for $p=2$. Moreover the results are consistent with results in [37]. For the special case, $p=1$, Theorem 2 reduces to Corollary 2.1 of [35].

Remark 2.

We note that, if for each $k\in \mathbb{Z}(1,T),f(k,·):\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying $f(k,t)t\ge 0$ for all $t\in \mathbb{R}\setminus \left\{0\right\}$, then

$$A_{+\infty }=\underset{t\to +\infty }{lim\; inf}\frac{{\displaystyle {\displaystyle \sum _{k=1}^T}F(k,t)}}{t^p},and{A}_{-\infty }=\underset{t\to -\infty }{lim\; inf}\frac{{\displaystyle {\displaystyle \sum _{k=1}^T}F(k,t)}}{{\left|t\right|}^p}.$$

Consequently, Theorem 1 immediately follows by Corollaries 1 and 2.

4. Two Examples

Example 1.

For $1\le p<2$, we consider the boundary value problem $(D_p^{\lambda ,f})$ with

$$f(k,t)={p\left|t\right|}^{p-1}\mathrm{sign}(t)\left(\frac{T+1}{T}+sin\left(\frac{1}{2T}{ln(|t|}^p+1)\right)+\frac{1}{2T}cos\left(\frac{1}{2T}{ln(|t|}^p+1)\right)\right),$$

(9)

for $k\in \mathbb{Z}(1,T)$, then

$$F(k,t)={\int }_0^{t}f(k,\tau )d\tau =\frac{T+1}{T}{\left|t\right|}^p+{(|t|}^p+1)sin\left(\frac{1}{2T}{ln(|t|}^p+1)\right),fort\in \mathbb{R}.$$

Since $f(k,t)\ge p{t}^{p-1}\left(\frac{T+1}{T}-1-\frac{1}{2T}\right)=\frac{p}{2T}{t}^{p-1}>0$, for $t>0$ and $f(k,0)=0$, we follow that for each fixed $k\in \mathbb{Z}(1,T)$, $F(k,t)$ is strictly monotone increasing on $[0,+\infty )$. One has $\underset{0\le s\le t}{max}F(k,s)=F(k,t)$, for each $t\ge 0$. Clearly,

$$A_{+\infty }=\underset{t\to +\infty }{lim\; inf}\frac{TF(k,t)}{t^p}=\underset{t\to +\infty }{lim\; inf}\frac{(T+1)t^p+T(t^p+1)sin(\frac{1}{2T}ln(t^p+1))}{t^p}=1,$$

and

$$B^{+\infty }=\underset{t\to +\infty }{lim\; sup}\frac{TF(k,t)}{t^p}=\underset{t\to +\infty }{lim\; sup}\frac{(T+1)t^p+T(t^p+1)sin(\frac{1}{2T}ln(t^p+1))}{t^p}=2T+1.$$

In view of $1\le p<2$ , we follow that $A_{+}\infty <\frac{2^{p-1}}{{(T+1)}^{\frac{p}{2}}}{B}^{+\infty }$. Applying to Theorem 2, problem $(D_p^{\lambda ,f})$ admits an unbounded sequence of positive solutions.

Let us consider another example.

Example 2.

Let $T=4,p=3$ andf be a function defined as follows

$$f(k,t)=3\left|t\right|t\left(\frac{5}{4}+sin(\frac{1}{8}{ln(|t|}^3+1))+\frac{1}{8}cos(\frac{1}{8}{ln(|t|}^3+1))\right),k\in \mathbb{Z}(1,4)$$

Then, for every $\lambda \in (\frac{2\sqrt{2}}{27},\frac{8}{75})$, the problem

$$\left\{\begin{array}{c}-▵\left({\varphi }_{3,c}\left(▵u(k-1)\right)\right)=\lambda f(k,u(k)),k\in \mathbb{Z}(1,4),\hfill \\ u(0)=u(5)=0,\hfill \end{array}\right.$$

(10)

Admits an unbounded sequence of positive solutions and an unbounded sequence of negative solutions. Indeed, $f(k,t)\ge 3t^2\left(\frac{5}{4}-1-\frac{1}{8}=\frac{3}{8}{t}^2\right)>0$, for $t>0$ and $f(k,0)=0.$

$$F(k,t)={\int }_0^{t}f(k,\tau )d\tau =\frac{5}{4}{\left|t\right|}^3+{(|t|}^3+1)sin(\frac{1}{8}{ln(|t|}^3+1)),fort\in \mathbb{R}.$$

Since $f(k,t)\ge 3t^2\left(\frac{5}{4}-1-\frac{1}{8}\right)=\frac{3}{8}{t}^2>0$, for $t>0$, we follow that for each fixed $k\in \mathbb{Z}(1,4)$, $F(k,t)$ is strictly monotone increasing on $[0,+\infty )$. Thus, $\underset{\left|s\right|\le t}{max}F(k,s)=F(k,t)$, for each $t\ge 0$. Obviously,

$$A_{\pm \infty }=\underset{t\to +\infty }{lim\; inf}\frac{4F(k,t)}{t^3}=\underset{t\to +\infty }{lim\; inf}\frac{5t^3+4(t^3+1)sin(\frac{1}{8}ln(t^3+1))}{t^3}=1,$$

and

$$B^{\pm \infty }=\underset{t\to +\infty }{lim\; sup}\frac{4F(k,t)}{t^3}=\underset{t\to +\infty }{lim\; sup}\frac{5t^3+4(t^3+1)sin(\frac{1}{8}ln(t^3+1))}{t^3}=9.$$

Through simple computation, $max\{A_{+\infty },A_{-\infty }\}<\frac{{(\sqrt{2})}^p}{{(T+1)}^{p-1}}min\{B^{+\infty },B^{-\infty }\}$ holds. Corollary 2 ensures our claim.

5. Conclusions

In this paper, we have discussed the Dirichlet boundary value problem of the difference equation with p-mean curvature operator. Some sufficient conditions are derived for the existence of sequences of constant-sign solutions to the problem. Two examples are given to show the effectiveness of our results.

To solve problem $(D_p^{\lambda ,f})$, we further develop the methods adopted in [23]. The approaches can be used for the boundary value problems of differential equations involving p-mean curvature operator. Therefore, our work has both theoretical and practical significance.