1 Introduction

Below N, Z and R denote the sets of all natural numbers, integers and real numbers respectively. k is a positive integer. For any a, b ∈Z, define Z(a)={a,a+1,…},Z(a,b)={a,a+1,…,b} when a<b. Δ is the forward difference operator Δu n =u n+1−u n , Δ2 u n =Δ(Δu n ). Besides, * denotes the transpose of a vector.

Recently, difference equations have attracted the interest of many researchers since they provided a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural networks, ecology, cybernetics, biological systems, optimal control, and population dynamics. These studies cover many of the branches of difference equations, such as stability, attractivity, periodicity, oscillation, and boundary value problems, see [8, 17–19, 23, 26, 27, 29, 34, 42] and the references therein.

The present paper considers the fourth-order nonlinear difference equation

$$ \Delta^2 \bigl(p_{n-1}\Delta^2u_{n-2} \bigr)-\Delta \bigl(q_n (\Delta u_{n-1} )^\delta \bigr) +r_nu_n^\delta=f(n,u_n),\quad n \in \mathbf{Z}(1,k), $$
(1.1)

with boundary value conditions

$$ \Delta u_{-1}=\Delta u_0=0,\quad\quad u_{k+1}=u_{k+2}=0, $$
(1.2)

where p n is nonzero and real valued for each n∈Z(0,k+1), {q n } n∈Z(1,k+1) and {r n } n∈Z(1,k) are real sequences, δ is the ratio of odd positive integers, f∈C(R 2,R).

We may think of (1.1) with (1.2) as being a discrete analogue of the following fourth-order nonlinear differential equation

$$ \bigl[p(t)u''(t) \bigr]''- \bigl[q(t)u'(t) \bigr]'=f\bigl(t,u(t)\bigr),\quad t\in [a,b], $$
(1.3)

with boundary value conditions

$$ u(a)=u'(a)=0,\quad\quad u(b)=u'(b)=0. $$
(1.4)

Equation (1.3) includes the following equation

$$ u^{(4)}(t)=f\bigl(t,u(t)\bigr),\quad t\in \textbf{R}, $$
(1.5)

which is used to describe the bending of an elastic beam; see, for example, [5, 16, 20, 22, 25, 39] and the references therein. Owing to its importance in physics, many methods are applied to study fourth-order boundary value problems by many authors.

In recent years, the study of boundary value problems for differential equations develops at relatively rapid rate. By using various methods and techniques, such as fixed point theory, topological degree theory, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in literatures, we refer to [2–4, 7, 21, 37]. And critical point theory is also an important tool to deal with problems on differential equations [11, 14, 15, 28, 33, 43]. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [17–19] and Shi et al. [35] have successfully proved the existence of periodic solutions of second-order nonlinear difference equations. We also refer to [40, 41] for the discrete boundary value problems. Compared to first-order or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention (see, for example, [9, 10, 13, 31, 32, 36, 38] and the references contained therein). Yan, Liu [38] in 1997 and Thandapani, Arockiasamy [36] in 2001 studied the following fourth-order difference equation of form,

$$ \numberwithin{equation} {section} \Delta^2 \bigl(p_{n} \Delta^2u_{n} \bigr)+f(n,u_n)=0,\quad n\in \mathbf{Z}. $$
(1.6)

The authors obtain criteria for the oscillation and nonoscillation of solutions for equation (1.6). In 2005, Cai, Yu and Guo [6] have obtained some criteria for the existence of periodic solutions of the fourth-order difference equation

$$ \numberwithin{equation} {section} \Delta^2 \bigl(p_{n-2} \Delta^2u_{n-2} \bigr)+f(n,u_n)=0,\quad n\in \mathbf{Z}. $$
(1.7)

In 1995, Peterson and Ridenhour considered the disconjugacy of equation (1.7) when p n ≡1 and f(n,u n )=q n u n (see [31]).

The boundary value problem (BVP) for determining the existence of solutions of difference equations has been a very active area of research in the last twenty years, and for surveys of recent results, we refer the reader to the monographs by Agarwal et al. [1, 12, 24, 29, 34]. We use the critical point theory to give some sufficient conditions for the nonexistence and existence of solutions for the BVP (1.1) with (1.2). We shall study the suplinear and sublinear cases. The main idea in this paper is to transfer the existence of the BVP (1.1) with (1.2) into the existence of the critical points of some functional. The proof is based on the notable Mountain Pass lemma in combination with variational technique. The purpose of this paper is two-folded. On one hand, we shall further demonstrate the powerfulness of critical point theory in the study of solutions for boundary value problems of difference equations. On the other hand, we shall complement existing ones. The motivation for the present work stems from the recent paper in [11].

Let

$$\begin{aligned} \bar{p} =&\max\bigl\{p_n:n\in\textbf{Z}(1,k+1)\bigr\},\quad\quad \underline{p}=\min\bigl\{p_n:n\in\textbf{Z}(1,k+1)\bigr\}, \\ \bar{q} =&\max\bigl\{q_n:n\in\textbf{Z}(2,k+1)\bigr\},\quad\quad \underline{q}=\min\bigl\{q_n:n\in\textbf{Z}(2,k+1)\bigr\}, \\ \bar{r} =&\max\bigl\{r_n:n\in\textbf{Z}(1,k)\bigr\},\quad\quad \underline{r}=\min\bigl\{r_n:n\in\textbf{Z}(1,k)\bigr\}. \end{aligned}$$

Our main results are as follows.

Theorem 1.1

Assume that the following hypotheses are satisfied:

(p):

for any n∈Z(1,k+1), p n <0;

(q):

for any n∈Z(2,k+1), q n ≤0;

(r):

for any n∈Z(1,k), r n ≤0;

(F 1):

there exists a functional F(n,v)∈C 1(Z×R,R) with F(0,⋅)=0 such that

$$\begin{aligned} \frac{\partial F(n,v)}{\partial v} =f(n,v),\quad \forall n\in\textbf{Z}(1,k); \end{aligned}$$
(F 2):

there exists a constant M 0>0 for all (n,v)∈Z(1,k)×R such that |f(n,v)|≤M 0.

Then the BVP (1.1) with (1.2) possesses at least one solution.

Remark 1.1

Assumption (F 2) implies that there exists a constant M 1>0 such that \((F'_{2})\) |F(n,v)|≤M 1+M 0|v|,∀(n,v)∈Z(1,k)×R.

Theorem 1.2

Suppose that (F 1) and the following hypotheses are satisfied:

(p′):

for any n∈Z(1,k+1), p n >0;

(q′):

for any n∈Z(2,k+1), q n ≥0;

(r′):

for any n∈Z(1,k), r n ≥0;

(F 3):

there exists a functional F(n,v)∈C 1(Z×R,R) such that

$$\begin{aligned} \lim_{v\rightarrow0}\frac{F(n,v)}{|v|}=0, \quad \forall n\in \textbf{Z}(1,k); \end{aligned}$$
(F 4):

there exists a constant β>max{2,δ+1} such that for any n∈Z(1,k),

$$\begin{aligned} 0<vf(n,v)<\beta F(n,v),\quad\forall v\neq 0. \end{aligned}$$

Then the BVP (1.1) with (1.2) possesses at least two nontrivial solutions.

Remark 1.2

Assumption (F 4) implies that there exist constants a 1>0 and a 2>0 such that \((F'_{4})\) F(n,v)>a 1|v|β−a 2,∀n∈Z(1,k).

Theorem 1.3

Suppose that (p′),(q′),(r′),(F 1) and the following assumption are satisfied:

(F 5):

there exist constants R>0 and 1<α<2 such that for n∈Z(1,k) and |v|≥R,

$$\begin{aligned} 0<f(n,v)\leq \alpha F(n,v). \end{aligned}$$

Then the BVP (1.1) with (1.2) possesses at least one solution.

Remark 1.3

Assumption (F 5) implies that for each n∈Z(1,k) there exist constants a 3>0 and a 4>0 such that

\((F'_{5})\) :

F(n,v)≤a 3|v|α+a 4,∀(n,v)∈Z(1,k)×R.

Theorem 1.4

Suppose that (p),(q),(r),(F 1) and the following assumption are satisfied:

(F 6):

vf(n,v)>0, for v≠0,∀n∈Z(1,k).

Then the BVP (1.1) with (1.2) has no nontrivial solutions.

Remark 1.4

In the existing literature, results on the nonexistence of solutions of discrete boundary value problems are very scarce. Hence, Theorem 1.4 complements existing ones.

The remainder of this paper is organized as follows. First, in Sect. 2, we shall establish the variational framework for the BVP (1.1) with (1.2) and transfer the problem of the existence of the BVP (1.1) with (1.2) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Sect. 3, we shall complete the proof of the results by using the critical point method. Finally, in Sect. 4, we shall give three examples to illustrate the main results.

About the basic knowledge for variational methods, we refer the reader to [28, 30, 33, 43].

2 Variational structure and some lemmas

In order to apply the critical point theory, we shall establish the corresponding variational framework for the BVP (1.1) with (1.2) and give some lemmas which will be of fundamental importance in proving our main results. First, we state some basic notations.

Let R k be the real Euclidean space with dimension k. Define the inner product on R k as follows:

$$ \langle u,v\rangle =\sum^{k}_{j=1}u_j v_j,\quad \forall u,v\in \textbf{R}^k, $$
(2.1)

by which the norm ∥⋅∥ can be induced by

$$ \|u\|= \Biggl(\sum^{k}_{j=1}u_j^2 \Biggr)^{\frac{1}{2}},\quad\forall u\in \textbf{R}^k. $$
(2.2)

On the other hand, we define the norm ∥⋅∥ r on R k as follows:

$$ \|u\|_r= \Biggl(\sum^{k}_{j=1}|u_j|^r \Biggr)^{\frac{1}{r}}, $$
(2.3)

for all u∈R k and r>1.

Since ∥u∥ r and ∥u∥2 are equivalent, there exist constants c 1,c 2 such that c 2≥c 1>0, and

$$ c_1\|u\|_2\leq\|u\|_r\leq c_2\|u \|_2,\quad\forall u\in \textbf{R}^k. $$
(2.4)

Clearly, ∥u∥=∥u∥2. For any u=(u 1,u 2,…,u k )∗∈R k, for the BVP (1.1) with (1.2), consider the functional J defined on R k as follows:

$$\begin{aligned} J(u) =&\frac{1}{2}\sum_{n=1}^{k}p_{n+1} \bigl(\Delta^2u_n \bigr)^2 +\frac{1}{\delta+1} \sum_{n=1}^{k}q_{n+1} (\Delta u_n )^{\delta+1} +\frac{1}{\delta+1}\sum _{n=1}^{k}r_nu_n^{\delta+1} \\ &{} -\sum_{n=1}^{k}F(n,u_n)+ \frac{1}{2}p_1 (\Delta u_1 )^2, \end{aligned}$$
(2.5)

where

$$\begin{aligned} \frac{\partial F(n,v)}{\partial v} =f(n,v),\quad\quad\Delta u_{-1}=\Delta u_0=0,\quad\quad u_{k+1}=u_{k+2}=0. \end{aligned}$$

Clearly, J∈C 1(R k,R) and for any \(u=\{u_{n}\}_{n=1}^{k}=(u_{1},u_{2},\ldots,u_{k})^{\ast}\), by using Δu −1=Δu 0=0,u k+1=u k+2=0, we can compute the partial derivative as

$$\begin{aligned} \frac{\partial J}{\partial u_n}=\Delta^2 \bigl(p_{n-1} \Delta^2u_{n-2} \bigr)-\Delta \bigl(q_n (\Delta u_{n-1} )^\delta \bigr) +r_nu_n^\delta-f(n,u_n), \quad\forall n\in \mathbf{Z}(1,k). \end{aligned}$$

Thus, u is a critical point of J on R k if and only if

$$\begin{aligned} \Delta^2 \bigl(p_{n-1}\Delta^2u_{n-2} \bigr)-\Delta \bigl(q_n (\Delta u_{n-1} )^\delta \bigr) +r_nu_n^\delta=f(n,u_n),\quad \forall n\in \mathbf{Z}(1,k). \end{aligned}$$

We reduce the existence of the BVP (1.1) with (1.2) to the existence of critical points of J on R k. That is, the functional J is just the variational framework of the BVP (1.1) with (1.2).

Let P and Q be the k×k matrices defined by

$$\begin{aligned} P =& \left (\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 6& -4& 1& 0& 0& \cdots & 0& 0& 0 \\ -4& 6& -4& 1& 0& \cdots & 0& 0& 0 \\ 1& -4& 6& -4& 1& \cdots & 0& 0& 0 \\ 0& 1& -4& 6& -4& \cdots & 0& 0& 0 \\ \vdots &\vdots &\vdots &\vdots &\vdots &\ddots &\vdots &\vdots &\vdots \\ 0& 0& 0& 0& 0& \cdots & 6& -4& 1 \\ 0& 0& 0& 0& 0& \cdots & -4& 6& -4 \\ 0& 0& 0& 0& 0& \cdots & 1& -4& 6 \\ \end{array} \right ), \\ Q =& \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 2& -1& 0& \cdots & 0& 0 \\ -1& 2& -1& \cdots & 0& 0 \\ 0& -1& 2& \cdots & 0& 0 \\ \vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\ 0& 0& 0& \cdots & 2& -1 \\ 0& 0& 0& \cdots & -1& 2\\ \end{array} \right ). \end{aligned}$$

Clearly, P and Q are positive definite. Let λ 1,λ 2,…,λ k be the eigenvalues of P, \(\tilde{\lambda}_{1},\tilde{\lambda}_{2},\ldots,\tilde{\lambda}_{k}\) be the eigenvalues of Q. Applying matrix theory, we know \(\lambda_{j}>0, \tilde{\lambda}_{j}>0, j=1,2,\ldots,k\). Without loss of generality, we may assume that

$$\begin{aligned} &0<\lambda_1\leq\lambda_2\leq\cdots\leq \lambda_{k}, \end{aligned}$$
(2.6)
$$\begin{aligned} & 0<\tilde{\lambda}_1\leq\tilde{\lambda}_2\leq\cdots\leq \tilde{\lambda}_{k}. \end{aligned}$$
(2.7)

Let E be a real Banach space, J∈C 1(E,R), i.e., J is a continuously Fréchet-differentiable functional defined on E. J is said to satisfy the Palais-Smale condition (P.S. condition for short) if any sequence {u (l)}⊂E for which {J(u (l))} is bounded and J′(u (l))→0(l→∞) possesses a convergent subsequence in E.

Let B ρ denote the open ball in E about 0 of radius ρ and let ∂B ρ denote its boundary.

Lemma 2.1

(Mountain Pass lemma [33])

Let E be a real Banach space and J∈C 1(E,R) satisfy the P.S. condition. If J(0)=0 and

(J 1):

there exist constants ρ, a>0 such that \(J|_{\partial B_{\rho}}\geq a\), and

(J 2):

there exists e∈E∖B ρ such that J(e)≤0.

Then J possesses a critical value c≥a given by

$$ c=\inf_{g\in \varGamma}\max_{s\in[0,1]} J\bigl(g(s)\bigr), $$
(2.8)

where

$$ \varGamma =\bigl\{g\in C\bigl([0,1],E\bigr)|g(0)=0, g(1)=e\bigr\}. $$
(2.9)

Lemma 2.2

Suppose that (p′),(q′),(r′),(F 1),(F 3) and (F 4) are satisfied. Then the functional J satisfies the P.S. condition.

Proof

Let u (l)∈R k,l∈Z(1) be such that {J(u (l))} is bounded. Then there exists a positive constant M 2 such that

$$\begin{aligned} -M_2\leq J \bigl(u^{(l)} \bigr)\leq M_2,\quad \forall l\in \textbf{N}. \end{aligned}$$

By \((F'_{4})\), we have

$$\begin{aligned} -M_2 \leq& J \bigl(u^{(l)} \bigr)\\ =&\frac{1}{2}\sum _{n=1}^{k}p_{n+1} \bigl( \Delta^2u_n^{(l)} \bigr)^2 + \frac{1}{\delta+1}\sum_{n=1}^{k}q_{n+1} \bigl(\Delta u_n^{(l)} \bigr)^{\delta+1} \\ &{}+\frac{1}{\delta+1}\sum_{n=1}^{k}r_n \bigl(u_n^{(l)} \bigr)^{\delta+1} -\sum _{n=1}^{k}F \bigl(n,u_n^{(l)} \bigr)+\frac{1}{2}p_1 \bigl(\Delta u_1^{(l)} \bigr)^2 \\ \leq& \frac{\bar{p}}{2}\sum_{n=1}^{k} \bigl(u_{n+2}^{(l)}-2u_{n+1}^{(l)}+u_n^{(l)} \bigr)^2 +\frac{\bar{q}c_2^{\delta+1}}{\delta+1} \Biggl\{ \Biggl[\sum _{n=1}^{k} \bigl(u_{n+1}^{(l)}-u_n^{(l)} \bigr)^2 \Biggr]^{\frac{1}{2}} \Biggr\}^{\delta+1} \\ &{}+\frac{\bar{r}c_2^{\delta+1}}{\delta+1}\bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1} -a_1\sum_{n=1}^{k}\bigl \vert u_n^{(l)}\bigr \vert ^\beta+a_2k+ \bar{p}\bigl \Vert u^{(l)}\bigr \Vert ^2 \\ \leq&\frac{\bar{p}}{2} \bigl(u^{(l)} \bigr)^\ast P u^{(l)} +\frac{\bar{q}c_2^{\delta+1}}{\delta+1} \bigl[ \bigl(u^{(l)} \bigr)^\ast Q u^{(l)} \bigr]^{\frac{\delta+1}{2}} +\frac{\bar{r}c_2^{\delta+1}}{\delta+1} \bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1} \\ &{}-a_1c_1^{\beta} \bigl \Vert u^{(l)}\bigr \Vert ^\beta+a_2k+\bar{p} \bigl \Vert u^{(l)}\bigr \Vert ^2 \\ \leq& \biggl(\frac{\lambda_k}{2}+1 \biggr)\bar{p}\bigl \Vert u^{(l)} \bigr \Vert ^2 +\frac{\bar{q}c_2^{\delta+1}\tilde{\lambda}_k^{\frac{\delta+1}{2}}}{\delta+1}\bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1} +\frac{\bar{r}c_2^{\delta+1}}{\delta+1}\bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1} \\ &{}-a_1c_1^{\beta}\bigl \Vert u^{(l)}\bigr \Vert ^\beta+a_2k, \end{aligned}$$

where \(u^{(l)}= (u^{(l)}_{1},u^{(l)}_{2},\ldots,u^{(l)}_{k} )^{\ast},\ u^{(l)}\in \textbf{R}^{k}\). That is,

$$\begin{aligned} &a_1c_1^{\beta}\bigl \Vert u^{(l)}\bigr \Vert ^\beta- \biggl(\frac{\lambda_k}{2}+1 \biggr)\bar{p}\bigl \Vert u^{(l)}\bigr \Vert ^2 -\frac{\bar{q}c_2^{\delta+1}\tilde{\lambda}_k^{\frac{\delta+1}{2}}}{\delta+1}\bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1} -\frac{\bar{r}c_2^{\delta+1}}{\delta+1}\bigl \Vert u^{(l)}\bigr \Vert ^{\delta+1}\\ &\quad\leq M_2+a_2k. \end{aligned}$$

Since β>max{2,δ+1}, there exists a constant M 3>0 such that

$$\begin{aligned} \bigl \Vert u^{(l)}\bigr \Vert \leq M_3,\quad\forall l\in \textbf{N}. \end{aligned}$$

Therefore, {u (l)} is bounded on R k. As a consequence, {u (l)} possesses a convergence subsequence in R k. Thus the P.S. condition is verified. □

3 Proof of the main results

In this section, we shall prove our main results by using the critical point theory.

Proof of Theorem 1.1

By \((F'_{2})\), for any u=(u 1,u 2,…,u k )∗∈R k, we have

$$\begin{aligned} J(u) =&\frac{1}{2}\sum_{n=1}^{k}p_{n+1} \bigl(\Delta^2u_n \bigr)^2 +\frac{1}{\delta+1} \sum_{n=1}^{k}q_{n+1} (\Delta u_n )^{\delta+1} +\frac{1}{\delta+1}\sum _{n=1}^{k}r_nu_n^{\delta+1} \\ &{} - \sum_{n=1}^{k}F(n,u_n)+ \frac{1}{2}p_1 (\Delta u_1 )^2 \\ \leq& \frac{\bar{p}}{2}\sum_{n=1}^{k} (u_{n+2}-2u_{n+1}+u_n )^2 +M_0\sum_{n=1}^{k}\vert u_n\vert +M_1k \\ \leq&\frac{\bar{p}}{2}u^\ast P u+M_0\sqrt{k}\|u \|+M_1k \\ \leq& \frac{\bar{p}\lambda_1}{2}\|u\|^2+M_0\sqrt{k}\|u \|+M_1k \rightarrow-\infty \end{aligned}$$

as ∥u∥→+∞.

The above inequality means that −J(u) is coercive. By the continuity of J(u), J attains its maximum at some point, and we denote it \(\check{u}\), that is,

$$\begin{aligned} J(\check{u})=\max \bigl\{J(u)|u\in\textbf{R}^k \bigr\}. \end{aligned}$$

Clearly, \(\check{u}\) is a critical point of the functional J. This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2

By (F 3), for any \(\epsilon=\frac{\underline{p}\lambda_{1}}{4}\) (λ 1 can be referred to (2.6)), there exists ρ>0, such that

$$\begin{aligned} |F(n,v)|\leq\frac{\underline{p}\lambda_1}{4}|v|,\quad\forall n\in\textbf{Z}(1,k), \end{aligned}$$

for |v|≤ρ.

For any u=(u 1,u 2,…,u k )∗∈R k and ∥u∥≤ρ, we have |u n |≤ρ, n∈Z(1,k).

For any n∈Z(1,k),

$$\begin{aligned} J(u) =&\frac{1}{2}\sum_{n=1}^{k}p_{n+1} \bigl(\Delta^2u_n \bigr)^2 +\frac{1}{\delta+1} \sum_{n=1}^{k}q_{n+1} (\Delta u_n )^{\delta+1} +\frac{1}{\delta+1}\sum _{n=1}^{k}r_nu_n^{\delta+1}\\ &{}- \sum_{n=1}^{k}F(n,u_n)+ \frac{1}{2}p_1 (\Delta u_1 )^2 \\ \geq&\frac{\underline{p}}{2}\sum_{n=1}^{k} (u_{n+2}-2u_{n+1}+u_n )^2 - \frac{\underline{p}\lambda_1}{4}\sum_{n=1}^{k}\vert u_n\vert \\ \geq&\frac{\underline{p}}{2}u^\ast P u-\frac{\underline{p}\lambda_1}{4}\|u \|^2 \\ \geq&\frac{\underline{p}\lambda_1}{2}\|u\|^2 -\frac{\underline{p}\lambda_1}{4}\|u \|^2 \\ =&\frac{\underline{p}\lambda_1}{4}\|u\|^2, \end{aligned}$$

where u=(u 1,u 2,…,u k )∗,u∈R k.

Take \(a=\frac{\underline{p}\lambda_{1}}{4}\rho^{2}>0\). Therefore,

$$\begin{aligned} J(u)\geq a>0,\quad\forall u\in \partial B_\rho. \end{aligned}$$

At the same time, we have also proved that there exist constants a>0 and ρ>0 such that \(J|_{\partial B_{\rho}}\geq a\). That is to say, J satisfies the condition (J 1) of the Mountain Pass lemma.

For our setting, clearly J(0)=0. In order to exploit the Mountain Pass lemma in critical point theory, we need to verify other conditions of the Mountain Pass lemma. By Lemma 2.2, J satisfies the P.S. condition. So it suffices to verify the condition (J 2).

From the proof of the P.S. condition, we know

$$\begin{aligned} J(u)\leq \biggl(\frac{\lambda_k}{2}+1 \biggr)\bar{p}\Vert u\Vert ^2 +\frac{\bar{q}c_2^{\delta+1}\tilde{\lambda}_k^{\frac{\delta+1}{2}}}{\delta+1}\Vert u\Vert ^{\delta+1} +\frac{\bar{r}c_2^{\delta+1}}{\delta+1}\Vert u \Vert ^{\delta+1}-a_1c_1^{\beta}\|u \|^\beta+a_2k. \end{aligned}$$

Since β>max{2,δ+1}, we can choose \(\bar{u}\) large enough to ensure that \(J(\bar{u})<0\).

By the Mountain Pass lemma, J possesses a critical value c≥a>0, where

$$\begin{aligned} c=\inf_{h\in \varGamma}\sup_{s\in [0,1]} J\bigl(h(s)\bigr), \end{aligned}$$

and

$$\begin{aligned} \varGamma =\bigl\{h\in C\bigl([0,1],\textbf{R}^k\bigr)\mid h(0)=0, h(1)=\bar{u}\bigr\}. \end{aligned}$$

Let \(\tilde{u}\in \textbf{R}^{k}\) be a critical point associated to the critical value c of J, i.e., \(J(\tilde{u})=c\). Similar to the proof of the P.S. condition, we know that there exists \(\hat{u}\in \textbf{R}^{k}\) such that

$$\begin{aligned} J(\hat{u})=c_{\max}=\max_{s\in [0,1]}J\bigl(h(s)\bigr). \end{aligned}$$

Clearly, \(\hat{u}\neq 0\). If \(\tilde{u}\neq \hat{u}\), then the conclusion of Theorem 1.2 holds. Otherwise, \(\tilde{u}=\hat{u}\). Then \(c=J(\tilde{u})=c_{\max}=\max_{s\in [0,1]}J(h(s))\). That is,

$$\begin{aligned} \sup_{u\in \textbf{R}^k} J(u)=\inf_{h\in \varGamma} \sup _{s\in [0,1]} J\bigl(h(s)\bigr). \end{aligned}$$

Therefore,

$$\begin{aligned} c_{\max}=\max_{s\in [0,1]}J\bigl(h(s)\bigr),\quad \forall h\in \varGamma. \end{aligned}$$

By the continuity of J(h(s)) with respect to s, J(0)=0 and \(J(\bar{u})<0\) imply that there exists s 0∈(0,1) such that

$$\begin{aligned} J \bigl(h (s_0 ) \bigr)=c_{\max}. \end{aligned}$$

Choose h 1,h 2∈Γ such that {h 1(s)∣s∈(0,1)}∩{h 1(s)∣s∈(0,1)} is empty, then there exists s 1,s 2∈(0,1) such that

$$\begin{aligned} J \bigl(h_1 (s_1 ) \bigr)= J \bigl(h_2 (s_2 ) \bigr)=c_{\max}. \end{aligned}$$

Thus, we get two different critical points of J on R k denoted by

$$\begin{aligned} u^1=h_1 (s_1 ), u^2=h_2 (s_2 ). \end{aligned}$$

The above argument implies that the BVP (1.1) with (1.2) possesses at least two nontrivial solutions. The proof of Theorem 1.2 is finished. □

Proof of Theorem 1.3

We only need to find at least one critical point of the functional J defined as in (2.5).

By \((F'_{5})\), for any u=(u 1,u 2,…,u k )∗∈R k, we have

$$\begin{aligned} J(u) =&\frac{1}{2}\sum_{n=1}^{k}p_{n+1} \bigl(\Delta^2u_n \bigr)^2 +\frac{1}{\delta+1} \sum_{n=1}^{k}q_{n+1} (\Delta u_n )^{\delta+1} +\frac{1}{\delta+1}\sum _{n=1}^{k}r_nu_n^{\delta+1}\\ &{}- \sum_{n=1}^{k}F(n,u_n)+ \frac{1}{2}p_1 (\Delta u_1 )^2 \\ \geq&\frac{\underline{p}}{2}\sum_{n=1}^{k} (u_{n+2}-2u_{n+1}+u_n )^2 -a_3\sum_{n=1}^{k}\vert u_n\vert ^\alpha-a_4k \\ =&\frac{\underline{p}}{2}u^\ast P u -a_3 \Biggl[ \Biggl(\sum _{n=1}^{k}\vert u_n\vert ^\alpha \Biggr)^{\frac{1}{\alpha}} \Biggr]^\alpha-a_4k \\ \geq&\frac{\underline{p}\lambda_1}{2}\|u\|^2 -a_3c_2^\alpha \Biggl[ \Biggl(\sum_{n=1}^{k}u_n^2 \Biggr)^{\frac{1}{2}} \Biggr]^\alpha-a_4k \\ \geq&\frac{\underline{p}\lambda_1}{2}\|u\|^2 -a_3c_2^\alpha \|u\|^\alpha-a_4k\rightarrow+\infty \end{aligned}$$

as ∥u∥→+∞.

By the continuity of J, we know from the above inequality that there exist lower bounds of values of the functional. And this means that J attains its minimal value at some point which is just the critical point of J with the finite norm. □

Proof of Theorem 1.4

Assume, for the sake of contradiction, that the BVP (1.1) with (1.2) has a nontrivial solution. Then J has a nonzero critical point u ⋆. Since

$$\begin{aligned} \frac{\partial J}{\partial u_n}=\Delta^2 \bigl(p_{n-1} \Delta^2u_{n-2} \bigr) -\Delta \bigl(q_n (\Delta u_{n-1} )^\delta \bigr) +r_nu_n^\delta-f(n,u_n), \end{aligned}$$

we get

$$\begin{aligned} &\sum_{n=1}^{k}f\bigl(n,u_n^\star \bigr)u_n^\star \\ &\quad=\sum_{n=1}^{k} \bigl[ \Delta^2 \bigl(p_{n-1}\Delta^2u_{n-2}^\star \bigr) -\Delta \bigl(q_n \bigl(\Delta u_{n-1}^\star \bigr)^\delta \bigr)+r_n \bigl(u_n^\star \bigr)^\delta \bigr]u_n^\star \\ &\quad =\sum_{n=1}^{k}p_{n+1} \bigl(\Delta^2u_n^\star \bigr)^2 + \sum_{n=1}^{k}q_{n+1} \bigl(\Delta u_n^\star \bigr)^{\delta+1}+\sum _{n=1}^{k}r_n \bigl(u_n^\star \bigr)^{\delta+1}+p_1 \bigl(\Delta u_1^\star \bigr)^2\leq0. \end{aligned}$$
(3.1)

On the other hand, it follows from (F 6) that

$$ \sum_{n=1}^{k}f\bigl(n,u_n^\star \bigr)u_n^\star>0. $$
(3.2)

This contradicts (3.1) and hence the proof is complete. □

4 Examples

As an application of Theorems 1.2, 1.3 and 1.4, we give four examples to illustrate our main results.

Example 4.1

For n∈Z(1,k), assume that

$$ \Delta^4u_{n-2}-\Delta \bigl(9^n (\Delta u_{n-1} )^\delta \bigr)+3^nu_n^\delta= \beta \varphi(n)u_n\vert u_n\vert ^{\beta-2}, $$
(4.1)

with boundary value conditions (1.2), δ is the ratio of odd positive integers, β>max{2,δ+1}, φ is continuously differentiable and φ(n)>0,n∈Z(1,k) with φ(0)=0. We have

$$\begin{aligned} p_{n}\equiv 1,\quad\quad q_{n}=9^n,\quad\quad r_{n}=3^n,\quad\quad f(n,v)=\beta \varphi(n)v|v|^{\beta-2} \end{aligned}$$

and

$$\begin{aligned} F(n,v)=\varphi(n)|v|^\beta. \end{aligned}$$

It is easy to verify all the assumptions of Theorem 1.2 are satisfied and then the BVP (4.1) with (1.2) possesses at least two nontrivial solutions.

Example 4.2

For n∈Z(1,k), assume that

$$ \Delta^2 \bigl(8^{n-1}\Delta^2u_{n-2} \bigr)-\Delta \bigl(6^n (\Delta u_{n-1} )^\delta \bigr)+5^nu_n^\delta=\alpha \psi(n)u_n \vert u_n\vert ^{\alpha-2}, $$
(4.2)

with boundary value conditions (1.2), δ is the ratio of odd positive integers, 1<α<2, ψ is continuously differentiable and ψ(n)>0, n∈Z(1,k) with ψ(0)=0. We have

$$\begin{aligned} p_{n}=8^n,\quad\quad q_{n}=6^n, \quad\quad r_{n}=5^n,\quad\quad f(n,v)=\alpha \psi(n)v|v|^{\alpha-2} \end{aligned}$$

and

$$\begin{aligned} F(n,v)=\psi(n)|v|^\alpha. \end{aligned}$$

It is easy to verify all the assumptions of Theorem 1.3 are satisfied and then the BVP (4.2) with (1.2) possesses at least one solution.

Example 4.3

For n∈Z(1,k), assume that

$$ -\Delta^4u_{n-2}+\Delta \bigl(9^n (\Delta u_{n-1} )^\delta \bigr)-8^nu_n^\delta =\frac{4}{3}u_n^{\frac{1}{3}}, $$
(4.3)

with boundary value conditions (1.2), δ is the ratio of odd positive integers. We have

$$\begin{aligned} p_{n}\equiv-1,\quad\quad q_{n}=-9^n,\quad\quad r_{n}=-8^n,\quad\quad f(n,v)=\frac{4}{3}v^{\frac{1}{3}} \end{aligned}$$

and

$$\begin{aligned} F(n,v)=v^{\frac{4}{3}}. \end{aligned}$$

It is easy to verify all the assumptions of Theorem 1.4 are satisfied and then the BVP (4.3) with (1.2) has no nontrivial solutions.

Example 4.4

For n∈Z(1,k), assume that

$$ \Delta^4u_{n-2}=4u_n\vert u_n \vert ^3, $$
(4.4)

with boundary value conditions (1.2). We have

$$\begin{aligned} &\delta=1,\quad\quad\beta=5,\quad\quad p_{n}\equiv 1,\quad\quad q_{n}=r_{n}\equiv0,\\ & f(n,v)=4v|v|^3, \quad\quad F(n,v)=\frac{4}{5}|v|^5. \end{aligned}$$

It is easy to verify all the assumptions of Theorem 1.2 are satisfied and then the BVP (4.4) with (1.2) possesses two nontrivial solutions \(u_{n}=\sin(\frac{\pi}{2}n)\) and \(u_{n}=\cos(\frac{\pi}{2}n)\).