Abstract

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically 𝑝-linear difference system with 𝑝-Laplacian Ξ”(|Δ𝑒(π‘›βˆ’1)|π‘βˆ’2Δ𝑒(π‘›βˆ’1))+βˆ‡[βˆ’πΎ(𝑛,𝑒(𝑛))+π‘Š(𝑛,𝑒(𝑛))]=0, where π‘βˆˆ(1,+∞), π‘›βˆˆβ„€, π‘’βˆˆβ„π‘, 𝐾,π‘ŠβˆΆβ„€Γ—β„π‘β†’β„ are not periodic in 𝑛, and W is asymptotically 𝑝-linear at infinity.

1. Introduction

Consider the following 𝑝-Laplacian difference system:Ξ”ξ‚€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’2[]Δ𝑒(π‘›βˆ’1)+βˆ‡βˆ’πΎ(𝑛,𝑒(𝑛))+π‘Š(𝑛,𝑒(𝑛))=0,π‘›βˆˆβ„€,(1.1) where Ξ” is the forward difference operator defined by Δ𝑒(𝑛)=𝑒(𝑛+1)βˆ’π‘’(𝑛), Ξ”2𝑒(𝑛)=Ξ”(Δ𝑒(𝑛)), π‘βˆˆ(1,+∞), π‘›βˆˆβ„€, π‘’βˆˆβ„π‘, 𝐾, π‘Š: ℀×ℝ𝑁→ℝ are not periodic in 𝑛, π‘Š is asymptotically 𝑝-linear at infinity, and 𝐾 and π‘Š are continuously differentiable in π‘₯. As usual, we say that a solution 𝑒(𝑛) of (1.1) is homoclinic (to 0) if 𝑒(𝑛)β†’0 as π‘›β†’Β±βˆž. In addition, if 𝑒(𝑛)β‰’0, then 𝑒(𝑛) is called a nontrivial homoclinic solution.

When 𝑝=2, (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:[]Μˆπ‘’(𝑑)+βˆ‡βˆ’πΎ(𝑑,𝑒(𝑑))+π‘Š(𝑑,𝑒(𝑑))=0,π‘‘βˆˆβ„.(1.2)

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by PoincarΓ© [1]. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. For the existence of homoclinic solutions of problem (1.2), one can refer to the papers [2–5].

Difference equations usually describe evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the (𝑛+1)th generation π‘₯(𝑛+1) is a function of the 𝑛th generation π‘₯(𝑛). In fact, difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books [6–8].

In some recent papers [9–20], the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Along this direction, several authors [21–28] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated by the above papers, we consider the existence of homoclinic orbits for problem (1.1) by using the variant version of Mountain Pass Theorem. Our result is new, which seems not to have been considered in the literature. Here is our main result.

Theorem 1.1. Suppose that 𝐾 and π‘Š satisfy the following conditions. (K1)There are two positive constants 𝑏1 and 𝑏2 such that 𝑏1||π‘₯|𝑝≀𝐾(𝑛,π‘₯)≀𝑏2||π‘₯|𝑝,βˆ€(𝑛,π‘₯)βˆˆβ„€Γ—β„π‘.(1.3)(K2)There is a positive constant 𝑏3 such that 𝑏3|π‘₯|𝑝||||≀(βˆ‡πΎ(𝑛,π‘₯),π‘₯)β‰€βˆ‡πΎ(𝑛,π‘₯)|π‘₯|≀𝑝𝐾(𝑛,π‘₯),βˆ€(𝑛,π‘₯)βˆˆβ„€Γ—β„π‘.(1.4)(W1)π‘Š(𝑛,0)=0, βˆ‡π‘Š(𝑛,π‘₯)=π‘œ(|π‘₯|π‘βˆ’1) as |π‘₯|β†’0 uniformly for π‘›βˆˆβ„€.(W2)There exists a constant 𝑅>0 such that ||||βˆ‡π‘Š(𝑛,π‘₯)|π‘₯|π‘βˆ’1≀𝑅,βˆ€π‘›βˆˆβ„€,π‘₯βˆˆβ„π‘.(1.5)(W3)There exists a function π‘‰βˆžβˆˆπ‘™βˆž(β„€,ℝ+) such that lim|π‘₯|β†’βˆž||βˆ‡π‘Š(𝑛,π‘₯)βˆ’π‘‰βˆž(𝑛)|π‘₯|π‘βˆ’2π‘₯|||π‘₯|π‘βˆ’1=0uniformlyforπ‘›βˆˆβ„€,infβ„€π‘‰βˆžξ€½(𝑛)>max1,𝑝𝑏2ξ€Ύ.(1.6)(W4)ξ‚‹π‘Š(𝑛,π‘₯)=(βˆ‡π‘Š(𝑛,π‘₯),π‘₯)βˆ’π‘π‘Š(𝑛,π‘₯), lim|π‘₯|β†’βˆžξ‚‹π‘Š(𝑛,π‘₯)=+∞uniformlyforπ‘›βˆˆβ„€,(1.7) and for any fixed 0<𝑐1<𝑐2<+∞, infnβˆˆβ„€,𝑐1≀|π‘₯|≀𝑐2ξ‚‹π‘Š(𝑛,π‘₯)|π‘₯|𝑝>0.(1.8)Then problem (1.1) has at least one nontrivial homoclinic solution.

Remark 1.2. The function π‘Š(𝑛,π‘₯) in this paper is asymptotically 𝑝-linear at infinity. The behavior of the gradient of π‘Š(𝑛,π‘₯) at infinity is like that of a function π‘‰βˆž(𝑛)|π‘₯|π‘βˆ’2π‘₯, where π‘‰βˆž(𝑛) is a real function but not a matrix function. To the best of our knowledge, similar results of this kind of 𝑝-Laplacian difference systems with asymptotically 𝑝-linear π‘Š(𝑛,π‘₯) at infinity cannot be found in the literature. From this point, our result is new.

2. Preliminaries

Let 𝑆={𝑒(𝑛)}π‘›βˆˆβ„€βˆΆπ‘’(𝑛)βˆˆβ„π‘ξ€Ύ,,π‘›βˆˆβ„€πΈ=π‘’βˆˆπ‘†βˆΆπ‘›βˆˆβ„€ξ€Ί||||Δ𝑒(π‘›βˆ’1)𝑝+||||𝑒(𝑛)𝑝,<+∞(2.1) and for π‘’βˆˆπΈ, let ‖𝑒‖=π‘›βˆˆβ„€ξ€Ί||||Δ𝑒(π‘›βˆ’1)𝑝+||||𝑒(𝑛)𝑝<+∞1/𝑝.(2.2) Then 𝐸 is a uniform convex Banach space with this norm. As usual, for 1≀𝑝<+∞, let 𝑙𝑝℀,ℝ𝑁=ξƒ―ξ“π‘’βˆˆπ‘†βˆΆπ‘›βˆˆβ„€||||𝑒(𝑛)𝑝<+∞,π‘™βˆžξ€·β„€,ℝ𝑁=ξ‚»π‘’βˆˆπ‘†βˆΆsupπ‘›βˆˆβ„€||||ξ‚Ό,𝑒(𝑛)<+∞(2.3) and their norms are given by ‖𝑒‖𝑙𝑝=ξƒ©ξ“π‘›βˆˆβ„€||||𝑒(𝑛)𝑝ξƒͺ1/𝑝,βˆ€π‘’βˆˆπ‘™π‘ξ€·β„€,ℝ𝑁,β€–π‘’β€–βˆžξ€½||𝑒||ξ€Ύ=sup(𝑛)βˆΆπ‘›βˆˆβ„€,βˆ€π‘’βˆˆπ‘™βˆžξ€·β„€,ℝ𝑁,(2.4) respectively.

For any π‘’βˆˆπΈ, let1πœ‘(𝑒)=π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’ξ“π‘›βˆˆβ„€[].βˆ’πΎ(𝑛,𝑒(𝑛))+π‘Š(𝑛,𝑒(𝑛))(2.5)

To prove our results, we need the following generalization of Lebesgue's dominated convergence theorem.

Lemma 2.1 (see [29]). Let {π‘“π‘˜(𝑑)} and {π‘”π‘˜(𝑑)} be two sequences of measurable functions on a measurable set 𝐴, and let ||π‘“π‘˜||(𝑑)β‰€π‘”π‘˜(𝑑),βˆ€π‘Ž.𝑒.π‘‘βˆˆπ΄.(2.6) If limπ‘˜β†’βˆžπ‘“π‘˜(𝑑)=𝑓(𝑑),limπ‘˜β†’βˆžπ‘”π‘˜(𝑑)=𝑔(𝑑),βˆ€π‘Ž.𝑒.π‘‘βˆˆπ΄,limπ‘˜β†’βˆžξ€œπ΄π‘”π‘˜ξ€œ(𝑑)𝑑𝑑=𝐴𝑔(𝑑)𝑑𝑑<+∞,(2.7) then limπ‘˜β†’βˆžξ€œπ΄π‘“π‘˜(ξ€œπ‘‘)𝑑𝑑=𝐴𝑓(𝑑)𝑑𝑑.(2.8)

Lemma 2.2. For π‘’βˆˆπΈ, β€–π‘’β€–βˆžβ‰€β€–π‘’β€–π‘™π‘β‰€2‖𝑒‖.(2.9)

Proof. Since π‘’βˆˆπΈ, it follows that lim|𝑛|β†’βˆž|𝑒(𝑛)|=0. Hence, there exists π‘›βˆ—βˆˆβ„€ such that β€–π‘’β€–βˆž=||π‘’ξ€·π‘›βˆ—ξ€Έ||=maxπ‘›βˆˆβ„€||𝑒||.(𝑛)(2.10) Hence, we have β€–π‘’β€–βˆžβ‰€ξƒ©ξ“π‘›βˆˆβ„€||||𝑒(𝑛)𝑝ξƒͺ1/𝑝=‖𝑒‖𝑙𝑝=ξƒ©ξ“π‘›βˆˆβ„€||||𝑒(𝑛)βˆ’π‘’(π‘›βˆ’1)+𝑒(π‘›βˆ’1)𝑝ξƒͺ1/π‘β‰€ξƒ©ξ“π‘›βˆˆβ„€ξ€·||||+||||𝑒(𝑛)βˆ’π‘’(π‘›βˆ’1)𝑒(π‘›βˆ’1)𝑝ξƒͺ1/𝑝≀2π‘ξ“π‘›βˆˆβ„€ξ€·||||𝑒(𝑛)βˆ’π‘’(π‘›βˆ’1)𝑝+||||𝑒(π‘›βˆ’1)𝑝ξƒͺ1/𝑝=2π‘›βˆˆβ„€ξ€·||||Δ𝑒(π‘›βˆ’1)𝑝+||||𝑒(π‘›βˆ’1)𝑝ξƒͺ1/𝑝=2π‘›βˆˆβ„€ξ€·||||Δ𝑒(π‘›βˆ’1)𝑝+||||𝑒(𝑛)𝑝ξƒͺ1/𝑝=2‖𝑒‖.(2.11)

Lemma 2.3. Suppose that (K1), (K2), and (W2) hold. If π‘’π‘˜β†’π‘’ in 𝐸, then βˆ‡πΎ(𝑛,π‘’π‘˜)β†’βˆ‡πΎ(𝑛,𝑒) and βˆ‡π‘Š(𝑛,π‘’π‘˜)β†’βˆ‡π‘Š(𝑛,𝑒) in π‘™π‘ξ…ž(ℝ,ℝ𝑁), where π‘ξ…ž>1 satisfies 1/𝑝+1/π‘ξ…ž=1.

Proof. From (K1) and (K2), we have ||||βˆ‡πΎ(𝑛,π‘₯)≀𝑝𝑏2|π‘₯|π‘βˆ’1,βˆ€(𝑛,π‘₯)βˆˆβ„€Γ—β„π‘.(2.12) Hence, from (2.12), we have ||ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ||(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛))π‘ξ…žβ‰€ξ‚ƒπ‘π‘2ξ‚€||π‘’π‘˜||(𝑛)π‘βˆ’1+||||𝑒(𝑛)π‘βˆ’1ξ‚ξ‚„π‘ξ…žβ‰€ξ‚ƒπ‘π‘22π‘βˆ’1||π‘’π‘˜(||𝑛)βˆ’π‘’(𝑛)π‘βˆ’1+𝑝𝑏2ξ€·1+2π‘βˆ’1ξ€Έ||||𝑒(𝑛)π‘βˆ’1ξ‚„π‘ξ…žβ‰€2π‘π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…ž||π‘’π‘˜||(𝑛)βˆ’π‘’(𝑛)𝑝+2π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žξ€·1+2π‘βˆ’1ξ€Έπ‘ξ…ž||||𝑒(𝑛)π‘βˆΆ=π‘”π‘˜(𝑛).(2.13) Moreover, since π‘’π‘˜β†’π‘’ in 𝑙𝑝(β„€,ℝ𝑁) and π‘’π‘˜(𝑛)→𝑒(𝑛) for almost every π‘›βˆˆβ„€, hence, limπ‘˜β†’βˆžπ‘”π‘˜(𝑛)=2π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žξ€·1+2π‘βˆ’1ξ€Έπ‘ξ…ž||||𝑒(𝑛)π‘βˆΆ=𝑔(𝑛),βˆ€a.e.π‘›βˆˆβ„€,limπ‘˜β†’βˆžξ“π‘›βˆˆβ„€π‘”π‘˜(𝑛)=limπ‘˜β†’βˆžξ“π‘›βˆˆβ„€ξ‚ƒ2π‘π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…ž||π‘’π‘˜||(𝑛)βˆ’π‘’(𝑛)𝑝+2π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žξ€·1+2π‘βˆ’1ξ€Έπ‘ξ…ž||||𝑒(𝑛)𝑝=2π‘π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žlimπ‘˜β†’βˆžξ“π‘›βˆˆβ„€||π‘’π‘˜||(𝑛)βˆ’π‘’(𝑛)𝑝+2π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žξ€·1+2π‘βˆ’1ξ€Έπ‘ξ…žξ“π‘›βˆˆβ„€||𝑒||(𝑛)𝑝=2π‘ξ…žξ€·π‘π‘2ξ€Έπ‘ξ…žξ€·1+2π‘βˆ’1ξ€Έπ‘ξ…žξ“π‘›βˆˆβ„€||||𝑒(𝑛)𝑝=ξ“π‘›βˆˆβ„€π‘”(𝑛)<+∞.(2.14) It follows from Lemma  2.1, (2.13), and the previous equations that limπ‘˜β†’βˆžξ“π‘›βˆˆβ„€||ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ||(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛))π‘ξ…ž=0.(2.15) This shows that βˆ‡πΎ(𝑛,π‘’π‘˜)β†’βˆ‡πΎ(𝑛,𝑒) in π‘™π‘ξ…ž(β„€,ℝ𝑁). By a similar proof, we can prove that βˆ‡π‘Š(𝑛,π‘’π‘˜)β†’βˆ‡π‘Š(𝑛,𝑒) in π‘™π‘ξ…ž(β„€,ℝ𝑁). The proof is complete.

Lemma 2.4. Under the conditions of Theorem 1.1, one has ξ“βŸ¨πœ‘β€²(𝑒),π‘£βŸ©=π‘›βˆˆβ„€ξ‚ƒ||||Δ𝑒(π‘›βˆ’1)π‘βˆ’2ξ‚„(Δ𝑒(π‘›βˆ’1),Δ𝑣(π‘›βˆ’1))+(βˆ‡πΎ(𝑛,𝑒(𝑛))βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),𝑣(𝑛))(2.16) for 𝑒,π‘£βˆˆπΈ, which yields that ξ“βŸ¨πœ‘β€²(𝑒),π‘’βŸ©=π‘›βˆˆβ„€ξ€Ί||||Δ𝑒(π‘›βˆ’1)𝑝.+(βˆ‡πΎ(𝑛,𝑒(𝑛)),𝑒(𝑛))βˆ’(βˆ‡π‘Š(𝑛,𝑒(𝑛)),𝑒(𝑛))(2.17) Moreover, πœ‘ is continuously FrΓ©chet-differential defined on 𝐸; that is, πœ‘βˆˆπΆ1(𝐸,ℝ) and any critical point 𝑒 of πœ‘ on 𝐸 is classical solution of (1.1) with 𝑒(±∞)=0.

Proof. Firstly, we show that πœ‘βˆΆπΈβ†’β„. Let π‘’βˆˆπΈ, by (2.9) and (K1), we have ξ“π‘›βˆˆβ„€ξ“πΎ(𝑛,𝑒(𝑛))β‰€π‘›βˆˆβ„€π‘2||||𝑒(𝑛)𝑝≀𝑏22𝑝‖𝑒‖𝑝.(2.18) By (W2), we get ||π‘Š||=||||ξ€œ(𝑛,π‘₯)10||||(βˆ‡π‘Š(𝑛,𝑠π‘₯),π‘₯)𝑑𝑠≀𝑅|π‘₯|𝑝,βˆ€(𝑛,π‘₯)βˆˆβ„€Γ—β„π‘.(2.19) Hence, from (2.9) and (2.19), we have |||||ξ“π‘›βˆˆβ„€|||||β‰€ξ“π‘Š(𝑛,𝑒(𝑛))π‘›βˆˆβ„€||||β‰€ξ“π‘Š(𝑛,𝑒(𝑛))π‘›βˆˆβ„€π‘…||||𝑒(𝑛)𝑝≀𝑅2𝑝‖𝑒‖𝑝.(2.20) It follows from (2.5), (2.18), and (2.20) that πœ‘βˆΆπΈβ†’β„. Next we prove that πœ‘βˆˆπΆ1(𝐸,ℝ). Rewrite πœ‘ as follows: πœ‘(𝑒)=πœ‘1(𝑒)+πœ‘2(𝑒)βˆ’πœ‘3(𝑒),(2.21) where πœ‘11(𝑒)∢=π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)𝑝,πœ‘2(𝑒)∢=π‘›βˆˆβ„€πΎ(𝑛,𝑒(𝑛)),πœ‘3(𝑒)∢=π‘›βˆˆβ„€π‘Š(𝑛,𝑒(𝑛)).(2.22) It is easy to check that πœ‘1∈𝐢1(𝐸,ℝ) and ξ«πœ‘ξ…ž1=(𝑒),π‘£π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’2(Δ𝑒(π‘›βˆ’1),Δ𝑣(π‘›βˆ’1)),βˆ€π‘’,π‘£βˆˆπΈ.(2.23) Next, we prove that πœ‘π‘–βˆˆπΆ1(𝐸,ℝ),𝑖=2,3, and ξ«πœ‘ξ…ž2(=𝑒),π‘£π‘›βˆˆβ„€(βˆ‡πΎ(𝑛,𝑒(𝑛)),𝑣(𝑛)),βˆ€π‘’,π‘£βˆˆπΈ,(2.24)ξ«πœ‘ξ…ž3=(𝑒),π‘£π‘›βˆˆβ„€(βˆ‡π‘Š(𝑛,𝑒(𝑛)),𝑣(𝑛)),βˆ€π‘’,π‘£βˆˆπΈ.(2.25) For any 𝑒,π‘£βˆˆπΈ and for any function πœƒβˆΆβ„β†’(0,1), by (K2), we have ξ“π‘›βˆˆβ„€max[]β„Žβˆˆ0,1||||(βˆ‡πΎ(𝑛,𝑒(𝑛)+πœƒ(𝑑)β„Žπ‘£(𝑛)),𝑣(𝑛))≀𝑝𝑏2ξ“π‘›βˆˆβ„€max[]β„Žβˆˆ0,1||||𝑒(𝑛)+πœƒ(𝑑)β„Žπ‘£(𝑛)π‘βˆ’1||||𝑣(𝑛)≀2π‘βˆ’1𝑝𝑏2ξ“π‘›βˆˆβ„€ξ‚€||||𝑒(𝑛)π‘βˆ’1+||||𝑣(𝑛)π‘βˆ’1||||𝑣(𝑛)≀2π‘βˆ’1𝑝𝑏2ξ‚ƒβ€–π‘’β€–π‘™π‘βˆ’1𝑝‖𝑣‖𝑙𝑝+‖𝑣‖𝑝𝑙𝑝<+∞.(2.26) Then by the previous equations and Lebesgue's dominated convergence theorem, we have ξ«πœ‘ξ…ž2(𝑒),𝑣=limβ„Žβ†’0+πœ‘2(𝑒+β„Žπ‘£)βˆ’πœ‘2(𝑒)β„Ž=limβ„Žβ†’0+1β„Žξ“π‘›βˆˆβ„€[]𝐾(𝑛,𝑒(𝑛)+β„Žπ‘£(𝑛))βˆ’πΎ(𝑛,𝑒(𝑛))=limβ„Žβ†’0+ξ“π‘›βˆˆβ„€=(βˆ‡πΎ(𝑑,𝑒(𝑛)+πœƒ(𝑑)β„Žπ‘£(𝑛)),𝑣(𝑛))π‘›βˆˆβ„€(βˆ‡πΎ(𝑛,𝑒(𝑛)),𝑣(𝑛)),βˆ€π‘’,π‘£βˆˆπΈ.(2.27) Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that πœ‘π‘–βˆˆπΆ1(𝐸,ℝ),𝑖=2,3. Let π‘’π‘˜β†’π‘’ in 𝐸; then by Lemma 2.3, we have ||ξ«πœ‘ξ…ž2ξ€·π‘’π‘˜ξ€Έβˆ’πœ‘ξ…ž2||=|||||(𝑒),π‘£π‘›βˆˆβ„€ξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έξ€Έ|||||≀(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),𝑣(𝑛)π‘›βˆˆβ„€||ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ||(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛))‖𝑣(𝑛)β‰€β€–π‘£β€–π‘›βˆˆβ„€||ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ||(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛))π‘ξ…žξƒ­1/π‘ξ…žβŸΆ0,π‘˜βŸΆβˆž,βˆ€π‘£βˆˆπΈ.(2.28) This shows that πœ‘2∈𝐢1(𝐸,ℝ). Similarly, we can prove that πœ‘3∈𝐢1(𝐸,ℝ). Furthermore, by a standard argument, it is easy to show that the critical points of πœ‘ in 𝐸 are classical solutions of (1.1) with 𝑒(±∞)=0. The proof is complete.

Lemma 2.5 (see [30]). Let 𝐸 be a real Banach space with its dual space πΈβˆ— and suppose that πœ‘βˆˆπΆ1(𝐸,ℝ) satisfies max{πœ‘(0),πœ‘(𝑒)}β‰€πœ‚0<πœ‚β‰€inf‖𝑒‖=πœŒπœ‘(𝑒),(2.29) for some πœ‚0<πœ‚, 𝜌>0, and π‘’βˆˆπΈ with ‖𝑒‖>𝜌. Let 𝑐β‰₯πœ‚ be characterized by 𝑐=infΞ₯βˆˆΞ“max0β‰€πœβ‰€1πœ‘(Ξ₯(𝜏)),(2.30) where Ξ“={Ξ₯∈𝐢([0,1],𝐸)∢Ξ₯(0)=0,Ξ₯(1)=𝑒} is the set of continuous paths joining 0 to 𝑒; then there exists {π‘’π‘˜}π‘˜βˆˆβ„•βŠ‚πΈ such that πœ‘ξ€·π‘’π‘˜ξ€Έξ€·β€–β€–π‘’βŸΆπ‘,1+π‘˜β€–β€–ξ€Έβ€–β€–πœ‘β€²(π‘’π‘˜)β€–β€–πΈβˆ—βŸΆ0asπ‘˜βŸΆβˆž.(2.31)

3. Proof of Theorem 1.1

Proof of Theorem 1.1. We divide the proof of Theorem 1.1 into three steps.
Step 1. From (W1), there exists 𝜌0>0 such that πΆβˆ‡π‘Š(𝑛,π‘₯)≀12𝑝|π‘₯|π‘βˆ’1,βˆ€π‘›βˆˆβ„€,|π‘₯|β‰€πœŒ0,(3.1) where 𝐢1=min{1/𝑝,𝑏1}. From (3.1), we have ξ€œπ‘Š(𝑛,π‘₯)=10β‰€ξ€œ(βˆ‡π‘Š(𝑛,𝑠π‘₯),π‘₯)𝑑𝑠10𝐢12𝑝|π‘₯|π‘π‘ π‘βˆ’1𝐢𝑑𝑠=1𝑝2𝑝|π‘₯|𝑝,βˆ€π‘›βˆˆβ„€,|π‘₯|β‰€πœŒ0.(3.2) Let 𝜌=𝜌0/2 and 𝑆={π‘’βˆˆπΈβˆ£β€–π‘’β€–=𝜌}; then from (2.9), we obtain β€–π‘’β€–βˆžβ‰€πœŒ0,‖𝑒‖𝑙𝑝≀2𝜌,βˆ€π‘’βˆˆπ‘†,(3.3) which together with (2.9), (3.2), and (K1) implies that 1πœ‘(𝑒)=π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’ξ“π‘›βˆˆβ„€[]β‰₯1βˆ’πΎ(𝑛,𝑒(𝑛))+π‘Š(𝑛,𝑒(𝑛))π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)𝑝+𝑏1ξ“π‘›βˆˆβ„€||||𝑒(𝑛)π‘βˆ’ξ“π‘›βˆˆβ„€πΆ1𝑝2𝑝||||𝑒(𝑛)𝑝1β‰₯min𝑝,𝑏1ξ‚Όβ€–π‘’β€–π‘βˆ’πΆ1𝑝2𝑝‖𝑒‖𝑝𝑙𝑝β‰₯𝐢1β€–π‘’β€–π‘βˆ’πΆ1𝑝‖𝑒‖𝑝=(π‘βˆ’1)𝐢1𝑝‖𝑒‖𝑝=𝛼1>0,π‘’βˆˆπ‘†.(3.4)Step 2. From (K1), we have 1πœ‘(𝑒)=π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’ξ“π‘›βˆˆβ„€[]≀1βˆ’πΎ(𝑛,𝑒(𝑛))+π‘Š(𝑛,𝑒(𝑛))π‘ξ“π‘›βˆˆβ„€||||Δ𝑒(π‘›βˆ’1)𝑝+𝑏2ξ“π‘›βˆˆβ„€||||𝑒(𝑛)π‘βˆ’ξ“π‘›βˆˆβ„€ξ‚»1π‘Š(𝑛,𝑒(𝑛))≀max𝑝,𝑏2ξ‚Όβ€–π‘’β€–π‘βˆ’ξ“π‘›βˆˆβ„€π‘Š(𝑛,𝑒(𝑛))≑𝐢2β€–π‘’β€–π‘βˆ’ξ“π‘›βˆˆβ„€π‘Š(𝑛,𝑒(𝑛)).(3.5) By (W2) and (W3), we get lim|π‘₯|β†’βˆžπ‘π‘Š(𝑛,π‘₯)|π‘₯|𝑝=π‘‰βˆž(𝑛)uniformlyforπ‘›βˆˆβ„€.(3.6) Let π‘Š(𝑛,π‘₯)=π‘π‘Š(𝑛,π‘₯)βˆ’π‘‰βˆž(𝑛)|π‘₯|𝑝; it follows from (W2), (W3), (2.19), and (3.6) that ξ‚΅π‘Š(𝑛,π‘₯)≀𝑝𝑅+supβ„€π‘‰βˆž(ξ‚Ά|𝑛)π‘₯|𝑝,βˆ€π‘₯βˆˆβ„π‘,lim|π‘₯|β†’βˆžπ‘Š(𝑛,π‘₯)|π‘₯|𝑝=0.(3.7) Define 𝐸1∢={𝑒(𝑛)=π‘₯π‘’βˆ’|𝑛|∢π‘₯βˆˆβ„π‘,π‘›βˆˆβ„€}βŠ‚πΈ with infβ„€π‘‰βˆžξ€½(𝑛)>max1,𝑝𝑏2||ξ€Ύξ€·1+1βˆ’π‘’|𝑛|βˆ’|π‘›βˆ’1|||𝑝.(3.8) By an easy calculation, we have ‖𝑒‖𝑝=ξ€·||1+1βˆ’π‘’|𝑛|βˆ’|π‘›βˆ’1|||𝑝‖𝑒‖𝑝𝑙𝑝.(3.9) In what follows, we prove that for some π‘’βˆˆπΈ1 with ‖𝑒‖=1, πœ‘(𝑠𝑒)β†’βˆ’βˆž as π‘ β†’βˆž. Otherwise, there exist a sequence {π‘ π‘˜} with π‘ π‘˜β†’βˆž as π‘˜β†’βˆž and a positive constant 𝐢3 such that πœ‘(π‘ π‘˜π‘’)β‰₯βˆ’πΆ3 for all π‘˜. From (3.5), we obtain βˆ’πΆ3π‘ π‘π‘˜β‰€πœ‘ξ€·π‘ π‘˜π‘’ξ€Έπ‘ π‘π‘˜β‰€πΆ2βˆ’1π‘ξ“π‘›βˆˆβ„€π‘Šξ€·π‘›,π‘ π‘˜ξ€Έπ‘’(𝑛)π‘ π‘π‘˜βˆ’1π‘ξ“π‘›βˆˆβ„€π‘‰βˆž||||(𝑛)𝑒(𝑛)𝑝≀𝐢2βˆ’1π‘ξ“π‘›βˆˆβ„€π‘Šξ€·π‘›,π‘ π‘˜π‘’ξ€Έ(𝑛)π‘ π‘π‘˜βˆ’1𝑝infβ„€π‘‰βˆž(𝑛)‖𝑒‖𝑝𝑙𝑝.(3.10) It follows from (3.7) that π‘Šξ€·π‘›,π‘ π‘˜ξ€Έπ‘’(𝑛)π‘ π‘π‘˜β‰€ξ‚΅π‘π‘…+supβ„€π‘‰βˆžξ‚Ά||||(𝑛)𝑒(𝑛)𝑝,π‘Šξ€·π‘›,π‘ π‘˜ξ€Έπ‘’(𝑛)||π‘ π‘˜||π‘βŸΆ0asπ‘˜βŸΆβˆž.(3.11) Hence, from Lebesgue's dominated theorem and (3.11), we have ξ“π‘›βˆˆβ„€π‘Šξ€·π‘›,𝑠𝑛𝑒(𝑛)||π‘ π‘˜||π‘βŸΆ0asπ‘˜βŸΆβˆž.(3.12) It follows from (3.8), (3.9), (3.10), and (3.12) that 𝐢0βŸ΅βˆ’3π‘ π‘π‘˜β‰€πΆ2βˆ’1𝑝||1+1βˆ’π‘’|𝑛|βˆ’|π‘›βˆ’1|||𝑝infβ„€π‘‰βˆž(𝑛)<0asπ‘˜βŸΆβˆž,(3.13) which is a contradiction. Hence, there exists π‘’βˆˆπΈ with ‖𝑒‖>𝜌 such that πœ‘(𝑒)≀0.Step 3. From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence {π‘’π‘˜}π‘˜βˆˆβ„•βŠ‚πΈ such that πœ‘ξ€·π‘’π‘˜ξ€Έξ€·β€–β€–π‘’βŸΆπ‘,1+π‘˜β€–β€–ξ€Έβ€–β€–πœ‘ξ…žξ€·π‘’π‘˜ξ€Έβ€–β€–πΈβˆ—βŸΆ0asπ‘˜βŸΆβˆž,(3.14) where πΈβˆ— is the dual space of 𝐸. In the following, we will prove that {π‘’π‘˜}π‘˜βˆˆβ„• is bounded in 𝐸. Otherwise, assume that β€–π‘’π‘˜β€–β†’βˆž as π‘˜β†’βˆž. Let π‘§π‘˜=π‘’π‘˜/β€–π‘’π‘˜β€–; we have β€–π‘§π‘˜β€–=1. It follows from (2.5), (2.16), (3.14), and (K2) that 𝐢4𝑒β‰₯π‘πœ‘π‘˜ξ€Έβˆ’ξ«ξ€·π‘’πœ‘β€²π‘˜ξ€Έ,π‘’π‘˜ξ¬=ξ“π‘›βˆˆβ„€ξ€·ξ€Ίξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έξ€·(𝑛)βˆ’π‘π‘Šπ‘›,π‘’π‘˜+(𝑛)ξ€Έξ€»π‘›βˆˆβ„€ξ€Ίξ€·π‘πΎπ‘›,π‘’π‘˜ξ€Έβˆ’ξ€·ξ€·(𝑛)βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜β‰₯(𝑛)ξ€Έξ€»π‘›βˆˆβ„€ξ€·ξ€Ίξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έξ€·(𝑛)βˆ’π‘π‘Šπ‘‘,π‘’π‘˜ξ“(𝑛)ξ€Έξ€»βˆΆ=π‘›βˆˆβ„€ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜ξ€Έ.(𝑛)(3.15) Set Ξ©π‘˜(𝛼,𝛽)={π‘›βˆˆβ„€βˆΆπ›Όβ‰€|π‘’π‘˜(𝑛)|≀𝛽} for 0<𝛼<𝛽. Then from (3.15), we have 𝐢4β‰₯ξ“π‘›βˆˆΞ©π‘˜(0,𝛼)ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜(ξ€Έ+𝑛)π‘›βˆˆΞ©π‘˜(𝛼,𝛽)ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜(ξ€Έ+𝑛)π‘›βˆˆΞ©π‘˜(𝛽,+∞)ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜(ξ€Έ.𝑛)(3.16) From (K1), (K2), and (3.14), we get π‘œξ«ξ€·π‘’(1)=πœ‘β€²π‘˜ξ€Έ,π‘’π‘˜ξ¬=ξ“π‘›βˆˆβ„€ξ€Ί||Ξ”π‘’π‘˜||(π‘›βˆ’1)𝑝+ξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έξ€·(𝑛)βˆ’βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜β‰₯(𝑛)ξ€Έξ€»π‘›βˆˆβ„€ξ€Ί||Ξ”π‘’π‘˜(||π‘›βˆ’1)𝑝+𝑏3||π‘’π‘˜(||𝑛)π‘βˆ’ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜(𝑛),π‘’π‘˜(𝑛)ξ€Έξ€»β‰₯min1,𝑏3ξ€Ύβ€–β€–π‘’π‘˜β€–β€–π‘βˆ’ξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)∢=𝐢5β€–β€–π‘’π‘˜β€–β€–π‘βˆ’ξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ=‖‖𝑒(𝑛)π‘˜β€–β€–π‘ξƒ©πΆ5βˆ’ξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)β€–β€–π‘’π‘˜β€–β€–π‘ξƒͺ,(3.17) which implies that limsupπ‘˜β†’βˆžξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)||π‘’π‘˜||(𝑛)𝑝||π‘§π‘˜||(𝑛)𝑝=limsupπ‘˜β†’βˆžξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)β€–β€–π‘’π‘˜β€–β€–π‘β‰₯𝐢5.(3.18) Let 0<πœ€<𝐢5/3. From (W1), there exists π›Όπœ€>0 such that ||||β‰€πœ€βˆ‡π‘Š(𝑛,π‘₯)2𝑝|π‘₯|π‘βˆ’1for|π‘₯|β‰€π›Όπœ€uniformlyforπ‘›βˆˆβ„€.(3.19) Since β€–π‘§π‘˜β€–=1, it follows from (2.9) and (3.19) that ξ“π‘›βˆˆΞ©π‘˜(0,π›Όπœ€)||ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ||(𝑛)||π‘’π‘˜||(𝑛)π‘βˆ’1||π‘§π‘˜||(𝑛)π‘β‰€ξ“π‘›βˆˆΞ©π‘˜(0,π›Όπœ€)πœ€2𝑝||π‘§π‘˜||(𝑛)π‘β‰€πœ€,βˆ€π‘˜βˆˆβ„•.(3.20) For 𝑠>0, let ξ‚†ξ‚‹β„Ž(𝑠)∢=infπ‘Š(𝑛,π‘₯)βˆ£π‘›βˆˆβ„€,π‘₯βˆˆβ„π‘with.|π‘₯|β‰₯𝑠(3.21) Thus, from (W4), we have β„Ž(𝑠)β†’+∞ as 𝑠→+∞, which together with (3.16) implies that ξ€·Ξ©measπ‘˜ξ€Έβ‰€πΆ(𝛽,+∞)6β„Ž(𝛽)⟢0,asπ›½βŸΆ+∞.(3.22) Hence, we can take π›½πœ€ sufficiently large such that ξ“π‘›βˆˆΞ©π‘˜ξ€·π›½πœ€ξ€Έ,+∞||π‘§π‘˜||(𝑛)𝑝<πœ€π‘….(3.23) The previous inequality and (W2) imply that ξ“π‘›βˆˆΞ©π‘˜ξ€·π›½πœ€ξ€Έ,+∞||ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ||(𝑛)||π‘’π‘˜||(𝑛)π‘βˆ’1||π‘§π‘˜||(𝑛)π‘ξ“β‰€π‘…π‘›βˆˆΞ©π‘˜ξ€·π›½πœ€ξ€Έ,+∞||π‘§π‘˜||(𝑛)𝑝<πœ€,βˆ€π‘˜βˆˆβ„•.(3.24) Next, for the previous 0<π›Όπœ€<π›½πœ€, let π‘πœ€ξƒ―ξ‚‹βˆΆ=infπ‘Š(𝑛,π‘₯)|π‘₯|π‘βˆΆπ‘›βˆˆβ„€,π‘₯βˆˆβ„π‘withπ›Όπœ€β‰€|π‘₯|β‰€π›½πœ€ξƒ°,π‘‘πœ€ξ‚»||||∢=maxβˆ‡π‘Š(𝑛,π‘₯)|π‘₯|π‘βˆ’1βˆΆπ‘›βˆˆβ„€,π‘₯βˆˆβ„π‘withπ›Όπœ€β‰€|π‘₯|β‰€π›½πœ€ξ‚Ό.(3.25) From (W4), we have π‘πœ€>0 and ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜(𝑛)β‰₯π‘πœ€||π‘’π‘˜(||𝑛)𝑝,βˆ€π‘›βˆˆΞ©π‘˜ξ€·π›Όπœ€,π›½πœ€ξ€Έ.(3.26) From (3.15) and (3.26), we get ξ“π‘›βˆˆΞ©π‘˜(π›Όπœ€,π›½πœ€)||π‘§π‘˜||(𝑛)𝑝=1β€–β€–π‘’π‘˜β€–β€–π‘ξ“π‘›βˆˆΞ©π‘˜(π›Όπœ€,π›½πœ€)||π‘’π‘˜||(𝑛)𝑝≀1β€–β€–π‘’π‘˜β€–β€–π‘ξ“π‘›βˆˆΞ©π‘˜ξ€·π›Όπœ€,π›½πœ€ξ€Έ1π‘πœ€ξ‚‹π‘Šξ€·π‘›,π‘’π‘˜ξ€Έβ‰€πΆ(𝑛)4π‘πœ€β€–β€–π‘’π‘˜β€–β€–π‘βŸΆ0asπ‘˜βŸΆβˆž,(3.27) which implies that ξ“π‘›βˆˆΞ©π‘˜ξ€·π›Όπœ€,π›½πœ€ξ€Έ||ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ||(𝑛)||π‘’π‘˜||(𝑛)π‘βˆ’1||π‘§π‘˜||(𝑛)π‘β‰€π‘‘πœ€ξ“π‘›βˆˆΞ©π‘˜ξ€·π›Όπœ€,π›½πœ€ξ€Έ||π‘§π‘˜||(𝑛)π‘βŸΆ0asπ‘˜βŸΆβˆž.(3.28) Therefore, there exists π‘˜0>0 such that ξ“π‘›βˆˆΞ©π‘˜ξ€·π›Όπœ€,π›½πœ€ξ€Έ||ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ||(𝑛)||π‘’π‘˜||(𝑛)π‘βˆ’1||π‘§π‘˜||(𝑛)π‘β‰€πœ€,βˆ€π‘˜β‰₯π‘˜0.(3.29) It follows from (3.20), (3.24), and (3.29) that ξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)||π‘’π‘˜||(𝑛)𝑝||π‘§π‘˜||(𝑛)π‘β‰€ξ“π‘›βˆˆβ„€||ξ€·βˆ‡π‘Šπ‘‘,π‘’π‘˜ξ€Έ||(𝑛)||π‘’π‘˜||(𝑛)π‘βˆ’1||π‘§π‘˜||(𝑛)𝑝<3πœ€<𝐢5,βˆ€π‘˜β‰₯π‘˜0,(3.30) which implies that limsupπ‘›β†’βˆžξ“π‘›βˆˆβ„€ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛),π‘’π‘˜ξ€Έ(𝑛)||π‘’π‘˜||(𝑛)𝑝||π‘§π‘˜||(𝑛)𝑝<𝐢5,(3.31) but this contradicts to (3.18). Hence, β€–π‘’π‘˜β€– is bounded in 𝐸.
Going to a subsequence if necessary, we may assume that there exists π‘’βˆˆπΈ such that π‘’π‘˜β‡€π‘’ as π‘˜β†’βˆž. In order to prove our theorem, it is sufficient to show that πœ‘β€²(𝑒)=0. For any π‘Žβˆˆβ„€ with π‘Ž>0, let πœ’π‘Ž(𝑑)=1 for π‘‘βˆˆβ„€[βˆ’π‘Ž,π‘Ž] and let πœ’π‘Ž(𝑑)=0 for π‘‘βˆˆβ„€(βˆ’βˆž,βˆ’π‘Ž)βˆͺβ„€(π‘Ž,∞). Then from (2.16), we have ξ«ξ€·π‘’πœ‘β€²π‘˜ξ€Έβˆ’πœ‘β€²(𝑒),πœ’π‘Žξ€·π‘’π‘˜=ξ“βˆ’π‘’ξ€Έξ¬[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Ž||Ξ”π‘’π‘˜||(π‘›βˆ’1)π‘βˆ’2ξ€·Ξ”π‘’π‘˜(π‘›βˆ’1),Ξ”π‘’π‘˜ξ€Έβˆ’ξ“(π‘›βˆ’1)βˆ’Ξ”π‘’(π‘›βˆ’1)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Ž||||Δ𝑒(π‘›βˆ’1)π‘βˆ’2Δ𝑒(π‘›βˆ’1),Ξ”π‘’π‘˜ξ€Έ+(π‘›βˆ’1)βˆ’Ξ”π‘’(π‘›βˆ’1)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έβˆ’ξ“(𝑛)βˆ’π‘’(𝑛)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜(𝑛)βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),π‘’π‘˜(ξ€Έβ‰₯‖‖𝑛)βˆ’π‘’(𝑛)Ξ”π‘’π‘˜β€–β€–π‘π‘™π‘β„€[]βˆ’π‘Ž,π‘Ž+‖Δ𝑒‖𝑝𝑙𝑝℀[]βˆ’π‘Ž,π‘Žβˆ’ξ“[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Ž||Ξ”π‘’π‘˜||(π‘›βˆ’1)π‘βˆ’1||||βˆ’ξ“Ξ”π‘’(π‘›βˆ’1)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Ž||||Δ𝑒(π‘›βˆ’1)π‘βˆ’1||Ξ”π‘’π‘˜||+(π‘›βˆ’1)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έβˆ’ξ“(𝑛)βˆ’π‘’(𝑛)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έ(𝑛)βˆ’π‘’(𝑛)β‰₯β€–Ξ”π‘’π‘˜β€–π‘π‘™π‘β„€[]βˆ’π‘Ž,π‘Ž+‖Δ𝑒‖𝑝𝑙𝑝℀[βˆ’π‘Ž,π‘Ž]βˆ’β€–Ξ”π‘’β€–π‘™π‘β„€[βˆ’π‘Ž,π‘Ž]β€–Ξ”π‘’π‘˜β€–π‘™π‘βˆ’1𝑝℀[βˆ’π‘Ž,π‘Ž]βˆ’β€–Ξ”π‘’π‘˜β€–π‘™π‘β„€[βˆ’π‘Ž,π‘Ž]β€–Ξ”π‘’β€–π‘™π‘βˆ’1𝑝℀[βˆ’π‘Ž,π‘Ž]+ξ“π‘›βˆˆβ„€[βˆ’π‘Ž,π‘Ž]ξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έβˆ’ξ“(𝑛)βˆ’π‘’(𝑛)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έ=ξ‚€β€–β€–(𝑛)βˆ’π‘’(𝑛)Ξ”π‘’π‘˜β€–β€–π‘™π‘βˆ’1𝑝℀[]βˆ’π‘Ž,π‘Žβˆ’β€–Ξ”π‘’β€–π‘™π‘βˆ’1𝑝℀[]βˆ’π‘Ž,π‘Žξ‚ξ€·β€–β€–Ξ”π‘’π‘˜β€–β€–π‘™π‘β„€[βˆ’π‘Ž,π‘Ž]βˆ’β€–Ξ”π‘’β€–π‘™π‘β„€[βˆ’π‘Ž,π‘Ž]ξ€Έ+[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έβˆ’ξ“(𝑛)βˆ’π‘’(𝑛)[]π‘›βˆˆβ„€βˆ’π‘Ž,π‘Žξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έ.(𝑛)βˆ’π‘’(𝑛)(3.32) Since πœ‘β€²(π‘’π‘˜)β†’0 as π‘˜β†’+∞ and π‘’π‘˜β‡€π‘’ in 𝐸, it follows from (3.14) that ξ«ξ€·π‘’πœ‘β€²π‘˜ξ€Έβˆ’πœ‘β€²(𝑒),πœ’π‘Žξ€·π‘’π‘˜βˆ’π‘’ξ€Έξ¬βŸΆ0asξ“π‘˜βŸΆβˆž,π‘›βˆˆβ„€[βˆ’π‘Ž,π‘Ž]ξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡πΎ(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έ(𝑛)βˆ’π‘’(𝑛)⟢0asξ“π‘˜βŸΆβˆž,π‘›βˆˆβ„€[βˆ’π‘Ž,π‘Ž]ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έ(𝑛)βˆ’βˆ‡π‘Š(𝑛,𝑒(𝑛)),π‘’π‘˜ξ€Έ(𝑛)βˆ’π‘’(𝑛)⟢0asπ‘˜βŸΆβˆž.(3.33) It follows from (3.32) and (3.33) that β€–Ξ”π‘’π‘˜β€–π‘™π‘β„€[βˆ’π‘Ž,π‘Ž]→‖Δ𝑒‖𝑙𝑝℀[βˆ’π‘Ž,π‘Ž] as π‘˜β†’+∞.
For any π‘€βˆˆπΆβˆž0(ℝ,ℝ𝑁), and assume that for some π΄βˆˆβ„€ with 𝐴>0, supp(𝑀)βŠ‚β„€[βˆ’π΄,𝐴]. Since limπ‘˜β†’βˆžΞ”π‘’π‘˜(π‘›βˆ’1)=Δ𝑒(π‘›βˆ’1),βˆ€a.e|||ξ‚€||.π‘›βˆˆβ„€,Ξ”π‘’π‘˜||(π‘›βˆ’1)π‘βˆ’2Ξ”π‘’π‘˜ξ‚|||≀(π‘›βˆ’1),Δ𝑀(π‘›βˆ’1)π‘βˆ’1𝑝||Ξ”π‘’π‘˜||(π‘›βˆ’1)𝑝+1𝑝||||Δ𝑀(π‘›βˆ’1)𝑝,βˆ€π‘›βˆˆβ„€,π‘˜=1,2,…,limπ‘˜β†’βˆžξ“[]π‘›βˆˆβ„€βˆ’π΄,π΄ξ‚Έπ‘βˆ’1𝑝||Ξ”π‘’π‘˜||(π‘›βˆ’1)𝑝+1𝑝||||Δ𝑀(π‘›βˆ’1)𝑝=π‘βˆ’1𝑝limπ‘˜β†’βˆžβ€–β€–Ξ”π‘’π‘˜β€–β€–π‘π‘™π‘β„€[]βˆ’π΄,𝐴+1𝑝‖Δ𝑀‖𝑝𝑙𝑝℀[]βˆ’π΄,𝐴=π‘βˆ’1𝑝‖Δ𝑒‖𝑝𝑙𝑝℀[]βˆ’π΄,𝐴+1𝑝‖Δ𝑀‖𝑝𝑙𝑝℀[]βˆ’π΄,𝐴=[]π‘›βˆˆβ„€βˆ’π΄,π΄ξ‚Έπ‘βˆ’1𝑝||||Δ𝑒(π‘›βˆ’1)𝑝+1𝑝||||Δ𝑀(π‘›βˆ’1)𝑝<+∞,(3.34) then, we have ξ“π‘›βˆˆβ„€[βˆ’π΄,𝐴]ξ‚€||Ξ”π‘’π‘˜||(π‘›βˆ’1)π‘βˆ’2Ξ”π‘’π‘˜ξ‚βŸΆξ“(π‘›βˆ’1),Δ𝑀(π‘›βˆ’1)π‘›βˆˆβ„€[βˆ’π΄,𝐴]ξ‚€||||Δ𝑒(π‘›βˆ’1)π‘βˆ’2Δ𝑒(π‘›βˆ’1),Δ𝑀(π‘›βˆ’1)(3.35) as π‘˜β†’βˆž. Noting that []π‘›βˆˆβ„€βˆ’π΄,π΄ξ€·ξ€·βˆ‡πΎπ‘›,π‘’π‘˜(ξ€Έξ€ΈβŸΆξ“π‘›),𝑀(𝑛)π‘›βˆˆβ„€[βˆ’π΄,𝐴](βˆ‡πΎ(𝑛,𝑒(𝑛)),𝑀(𝑛))asξ“π‘˜βŸΆβˆž,π‘›βˆˆβ„€[βˆ’π΄,𝐴]ξ€·ξ€·βˆ‡π‘Šπ‘›,π‘’π‘˜ξ€Έξ€ΈβŸΆξ“(𝑛),𝑀(𝑛)π‘›βˆˆβ„€[βˆ’π΄,𝐴](βˆ‡π‘Š(𝑛,𝑒(𝑛)),𝑀(𝑛))asπ‘˜βŸΆβˆž.(3.36) Hence, we have βŸ¨πœ‘β€²(𝑒),π‘€βŸ©=limπ‘˜β†’βˆžξ«ξ€·π‘’πœ‘β€²π‘˜ξ€Έξ¬,𝑀=0,(3.37) which implies that πœ‘β€²(𝑒)=0; that is, 𝑒 is a critical point of πœ‘. From (K1) and (W1), we know that 𝑒≠0. In fact, if 𝑒=0, we have from (2.5), (K1), and (W1) that πœ‘(𝑒)=0. On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that πœ‘(𝑒)=𝑐>0. This is a contradiction. The proof of Theorem 1.1 is complete.

4. An Example

Example 4.1. In problem (1.1), let 𝑝=3/2, and ξ‚΅1𝐾(𝑛,π‘₯)=1+|π‘₯|3/2ξ‚Ά+1|π‘₯|3/2,π‘Š(𝑛,π‘₯)=π‘Ž(𝑛)|π‘₯|3/2ξ‚΅11βˆ’(ln(𝑒+|π‘₯|))1/2ξ‚Ά,(4.1) where π‘Žβˆˆπ‘™βˆž(β„€,ℝ+) with infβ„€π‘Ž(𝑛)>3. One can easily check that 𝐾 satisfies conditions (K1) and (K2) with 𝑏1=1, 𝑏2=2, and 𝑏3=3/2. An easy computation shows that 3βˆ‡π‘Š(𝑛,π‘₯)=2π‘Ž(𝑛)|π‘₯|βˆ’1/2π‘₯ξ‚΅11βˆ’(ln(𝑒+|π‘₯|))1/2ξ‚Ά+π‘Ž(𝑛)|π‘₯|1/2π‘₯2(𝑒+|π‘₯|)(ln(𝑒+|π‘₯|))3/2,3(βˆ‡π‘Š(𝑛,π‘₯),π‘₯)βˆ’2π‘Š(𝑛,π‘₯)=π‘Ž(𝑛)|π‘₯|5/22(𝑒+|π‘₯|)(ln(𝑒+|π‘₯|))3/2.(4.2) Then it is easy to check that π‘Š satisfies (W1)–(W4). Hence, 𝐾(𝑛,π‘₯) and π‘Š(𝑛,π‘₯) satisfy all the conditions of Theorem 1.1 and then problem (1.1) has at least one nontrivial homoclinic solution.

Acknowledgment

This work is partially supported by the NNSF (no. 10771215) of China.