In this section, we give our main results. Let

$$ mu =limsup_{xrightarrow +infty} frac{sum_{k=1}^{T}F(k,x)}{x^{2}} $$

(3.1)

and

$$ p_{*}=min bigl{ p_{k},kin mathbb{Z}(1,T)bigr} ,qquad p^{*}=max bigl{ p_{k},kin mathbb{Z}(1,T)bigr} . $$

We have the following result.

### Theorem 3.1

*Suppose that there are two real sequences* ({omega _{n}}), ({c_{n}}) *with* (omega _{n}>0) *and* (lim_{nrightarrow +infty}omega _{n}=+infty ) *such that*

$$ frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})c_{n}^{2}< frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}quad textit{for }nin mathbb{Z}(1) $$

(3.2)

*and*

$$ rho < frac{2mu}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.3)

*where*

$$ rho =liminf_{nrightarrow infty} frac{sum_{k=1}^{T}max_{|x|leq omega _{n}}F(k,x) -sum_{k=1}^{T}F(k,c_{n})}{ frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}-frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})c_{n}^{2}}. $$

*Then*, *for each* (lambda in (frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2mu}, frac{1}{rho} )), *problem* (1.1) *with* (1.2) *admits an unbounded sequence of solutions*.

### Proof

It is obvious that

$$ lim_{ Vert u Vert rightarrow +infty} Phi (u)=lim_{ Vert u Vert rightarrow + infty} frac{1}{2} sum_{k=0}^{T+1}p_{k-1} bigl(Delta ^{2}u_{k-1}bigr)^{2} geq lim _{ Vert u Vert rightarrow +infty}frac{p_{*}}{2} Vert u Vert ^{2} =+ infty , $$

which means that (Phi (u)) is coercive.

Define

$$ r_{n}=frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}. $$

If (uin E) and (Phi (u)< r_{n}), then we have the following inequality:

$$ frac{1}{2}p_{*}sum_{k=0}^{T+1} bigl(Delta ^{2}u_{k-1}bigr)^{2}< r_{n}. $$

Considering Lemma 2.2, for any (kin mathbb{Z}(1,T)), we have

$$ vert u_{k} vert ^{2}leq frac{(T+1)^{2}(T+3) }{32} sum_{k=0}^{T+1}bigl( Delta ^{2}u_{k-1}bigr)^{2}< omega _{n}^{2}. $$

Furthermore, according to the definition of *ϕ*, we have

$$ phi (r_{n})leq inf_{uin Phi ^{-1}(-infty ,r_{n})} frac{sum_{k=1}^{T}max_{|x|leq omega _{n}}F(k,x)-sum_{k=1}^{T}F(k,u_{k})}{ frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}-Phi (u)}. $$

(3.4)

For any (nin mathbb{Z}(1)), take ((q_{n})_{k}=c_{n}) for (kin mathbb{Z}(1,T)) and ((q_{n})_{-1}=(q_{n})_{0}=(q_{n})_{T}=(q_{n})_{T+1}=0), then (q_{n}in E) and

$$ Phi (q_{n})=frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})c_{n}^{2} leq r_{n} $$

by exploiting (3.2). Therefore, from (3.4), we have

$$ begin{aligned} phi (r_{n})&leq frac{sum _{k=1}^{T}max _{|x|leq omega _{n}}F(k,x) -sum _{k=1}^{T}F(k,(q_{n})_{k})}{ frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}-Phi (q_{n})} \ &= frac{sum _{k=1}^{T}max _{|x|leq omega _{n}}F(k,x) -sum _{k=1}^{T}F(k,c_{n})}{ frac{16p_{*}omega _{n}^{2}}{(T+1)^{2}(T+3)}-frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})c_{n}^{2}}. end{aligned} $$

Moreover, combining (3.3), it is clear that (alpha leq liminf_{nrightarrow +infty}phi (r_{n})leq rho <+ infty ).

We assert that (J_{lambda}) is unbounded from below. In fact, when (mu <+infty ), since

$$ 2lambda mu >p_{-1}+p_{0}+p_{T-1}+p_{T}, $$

there exists (varepsilon _{0}>0) such that

$$ 2lambda (mu -varepsilon _{0})>p_{-1}+p_{0}+p_{T-1}+p_{T}. $$

From (3.1), we know that there exists a positive sequence ({a_{n}} ) with (lim_{nrightarrow +infty}a_{n}=+infty ) such that

$$ sum_{k=1}^{T}F(k,a_{n}) geq (mu -varepsilon _{0})a_{n}^{2}. $$

For each (nin mathbb{Z}(1)), define (upsilon _{n}in E) with ((upsilon _{n})_{k}=a_{n}) for (kin mathbb{Z}(1,T)), then we have the following inequality:

$$ begin{aligned}[b] J_{lambda}(upsilon _{n})&=frac{1}{2}sum _{k=0}^{T+1}p_{k-1} bigl( Delta ^{2}(upsilon _{n})_{k-1} bigr)^{2} -lambda sum _{k=1}^{T} F bigl(k,(upsilon _{n})_{k}bigr) \ &leq frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})a_{n}^{2}- lambda (mu – varepsilon _{0})a_{n}^{2} \ &=frac{1}{2} bigl(p_{-1}+p_{0}+p_{T-1}+p_{T}-2 lambda (mu – varepsilon _{0}) bigr)a_{n}^{2}. end{aligned} $$

(3.5)

The above inequality implies (lim_{nrightarrow +infty}J_{lambda}(upsilon _{n})=-infty ). If (mu =+infty ), it can be seen that there is a sequence of positive number ({bar{a}_{n}}) with (lim_{nrightarrow +infty}bar{a}_{n}=+infty ) such that

$$ sum_{k=1}^{T}F(k,bar{a}_{n}) geq frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{lambda}bar{a}_{n}^{2} $$

from the definition of *μ*. Define (bar{upsilon}_{n}in E) as ((bar{upsilon}_{n})_{k}=bar{a}_{n}) for (kin mathbb{Z}(1,T)), then

$$ begin{aligned}[b] J_{lambda}(bar{ upsilon}_{n})&=frac{1}{2}sum _{k=0}^{T+1}p_{k-1}bigl(Delta ^{2}( bar{upsilon}_{n})_{k-1}bigr)^{2} -lambda sum _{k=1}^{T} Fbigl(k,(bar{ upsilon}_{n})_{k}bigr) \ &leq -frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T}) bar{a}_{n}^{2} rightarrow -infty quad text{as }n rightarrow +infty .end{aligned} $$

(3.6)

By combining (3.5) with (3.6), we can conclude that condition ((I_{1})) of Lemma 2.1 does not hold. Therefore, the functional (J_{lambda}) has a sequence of critical points with (lim_{nrightarrow +infty}Phi (u_{n})=+infty ), which means that the problem (1.1) with (1.2) admits an unbounded sequence of solutions. □

### Corollary 3.2

*If there is a sequence of positive numbers* ({tilde{omega}_{n}}) *with* (tilde{omega}_{n}rightarrow +infty ) *as* (nrightarrow +infty ) *such that*

$$ tilde{rho}< frac{2mu}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.7)

*where*

$$ tilde{rho}=liminf_{nrightarrow infty} frac{(T+1)^{2}(T+3)sum_{k=1}^{T}max_{|x|leq tilde{omega}_{n}}F(k,x) }{16p_{*}tilde{omega}_{n}^{2}},$$

*then*, *for each* (lambda in (frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2mu}, frac{1}{tilde{rho}} )), *problem* (1.1) *with* (1.2) *admits an unbounded sequence of nontrivial solutions*.

### Proof

Taking (c_{n}=0) for all (nin mathbb{Z}(1)), it can be easily proved by Theorem 3.1. □

In particular, if the nonlinear function *f* in (1.1) with the form (f(k,u)=a_{k}g(u)), where (a_{k}>0) for (kin mathbb{Z}(1,T)), and (p_{k}equiv 1) for (kin mathbb{Z}(-1,T)). Then (1.1) reads

$$ triangle ^{2} bigl( p_{k-2} triangle ^{2} u_{k-2} bigr)= lambda a_{k}g(u_{k}), quad k in mathbb{Z}(1,T). $$

(3.8)

Define

$$ bar{mu}=limsup_{xrightarrow +infty}frac{bar{G}(x)}{x^{2}}, $$

where

$$ bar{G}(x)= int _{0}^{x}g(s),ds. $$

Then we have the following.

### Corollary 3.3

*Suppose that there are two real sequences* ({bar{omega}_{n}}), ({bar{c}_{n}}) *with* (bar{omega}_{n}>0) *and* (lim_{nrightarrow +infty}bar{omega}_{n}=+infty ) *such that*

$$ frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T}) bar{c}_{n}^{2}< frac{16p_{*}bar{omega}_{n}^{2}}{(T+1)^{2}(T+3)}quad textit{for }n in mathbb{Z}(1) $$

(3.9)

*and*

$$ bar{rho}< frac{2bar{mu}}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.10)

*where*

$$ bar{rho}=liminf_{nrightarrow infty} frac{max_{|x|leq bar{omega}_{n}}bar{G}(x) -bar{G}(bar{c}_{n})}{ frac{16p_{*}bar{omega}_{n}^{2}}{(T+1)^{2}(T+3)}-frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})bar{c}_{n}^{2}}. $$

*Then*, *for each* (lambda in frac{1}{sum_{k=1}^{T}a_{k}} (frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2bar{mu}} ,frac{1}{bar{rho}})), *problem* (3.8) *with* (1.2) *admits an unbounded sequence of nontrivial solutions*.

Now, we discuss the existence of infinitely many solutions to the boundary value problem (1.1) with (1.2) by using the oscillatory behavior of the nonlinear function at the origin.

### Theorem 3.4

*Suppose that there are two real sequences* ({z_{n}}) *and* ({bar{z}_{n}}), *where* (bar{z}_{n}>0) *and* (lim_{nrightarrow +infty}bar{z}_{n}=0), *such that*

$$ frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})z_{n}^{2}< frac{16p_{*}bar{z}_{n}^{2}}{(T+1)^{2}(T+3)}~{mathrm{for}}~nin mathbb{Z}(1) $$

(3.11)

*and*

$$ varrho < frac{2mu}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.12)

*where*

$$ varrho =liminf_{nrightarrow infty} frac{sum_{k=1}^{T}max_{|x|leq bar{z}_{n}}F(k,x) -sum_{k=1}^{T}F(k,z_{n})}{ frac{16p_{*}bar{z}_{n}^{2}}{(T+1)^{2}(T+3)}-frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})z_{n}^{2}}. $$

*Then*, *for each* (lambda in ( frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2mu},frac{1}{varrho}) ), *problem* (1.1) *with* (1.2) *has a sequence of nontrivial solutions that converges to* 0.

The proof of Theorem 3.4 is similar to that of Theorem 3.1, so we omit it.

### Corollary 3.5

*Suppose that there is a sequence* ({tilde{z}_{n}}) *where* (tilde{z}_{n}>0) *and* (lim_{nrightarrow +infty}tilde{z}_{n}=0) *such that*

$$ bar{varrho}< frac{2mu}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.13)

*where*

$$ bar{varrho}=liminf_{nrightarrow infty} frac{(T+1)^{2}(T+3) sum_{k=1}^{T}max_{|x|leq tilde{z}_{n}}F(k,x)}{ 16p_{*}tilde{z}_{n}^{2}}.$$

*Then*, *for each* (lambda in (frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2mu}, frac{1}{bar{varrho}} )), *problem* (1.1) *with* (1.2) *has a sequence of nontrivial solutions that converges to* 0.

Considering the boundary value problem (3.8) with (1.2), we have the following result when the nonlinear function *g* oscillates at the origin.

### Corollary 3.6

*Suppose there are two real sequences* ({b_{n}}), ({bar{b}_{n}}) *with* (bar{b}_{n}>0) *and* (lim_{nrightarrow +infty}bar{b}_{n}=0) *such that*

$$ frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})b_{n}^{2}< frac{16p_{*}bar{b}_{n}^{2}}{(T+1)^{2}(T+3)}quad textit{for } nin mathbb{Z}(1) $$

(3.14)

*and*

$$ {sigma}< frac{2bar{mu}}{p_{-1}+p_{0}+p_{T-1}+p_{T}}, $$

(3.15)

*where*

$$ {sigma}=liminf_{nrightarrow infty} frac{max_{|x|leq bar{b}_{n}}bar{G}(x) -bar{G}(b_{n})}{ frac{16p_{*}bar{b}_{n}^{2}}{(T+1)^{2}(T+3)}-frac{1}{2}(p_{-1}+p_{0}+p_{T-1}+p_{T})b_{n}^{2}}. $$

*Then*, *for each* (lambda in frac{1}{sum_{k=1}^{T}a_{k}} ( frac{p_{-1}+p_{0}+p_{T-1}+p_{T}}{2bar{mu}} ,frac{1}{{sigma}} )), *problem* (3.8) *with* (1.2) *admits a sequence of nontrivial solutions that converges to* 0.

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