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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