Existence of Homoclinic Orbits for a Class of Asymptotically 𝑝-Linear Difference Systems with 𝑝-Laplacian

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 Δ(|Δ𝑢(𝑛−1)|^𝑝−2Δ𝑢(𝑛−1))+∇[−𝐾(𝑛,𝑢(𝑛))+𝑊(𝑛,𝑢(𝑛))]=0, where 𝑝∈(1,+∞), 𝑛∈ℤ, 𝑢∈ℝ^𝑁, 𝐾,𝑊∶ℤ×ℝ^𝑁→ℝ are not periodic in 𝑛, and W is asymptotically 𝑝-linear at infinity.

1. Introduction

Consider the following 𝑝-Laplacian difference system:Δ||Δ𝑢(𝑛−1)||^𝑝−2Δ𝑢(𝑛−1)+∇[−𝐾(𝑛,𝑢(𝑛))+𝑊(𝑛,𝑢(𝑛))]=0,𝑛∈ℤ,(1.1) where Δ is the forward difference operator defined by Δ𝑢(𝑛)=𝑢(𝑛+1)−𝑢(𝑛), Δ²𝑢(𝑛)=Δ(Δ𝑢(𝑛)), 𝑝∈(1,+∞), 𝑛∈ℤ, 𝑢∈ℝ^𝑁, 𝐾, 𝑊: ℤ×ℝ^𝑁→ℝ are not periodic in 𝑛, 𝑊 is asymptotically 𝑝-linear at infinity, and 𝐾 and 𝑊 are continuously differentiable in 𝑥. As usual, we say that a solution 𝑢(𝑛) of (1.1) is homoclinic (to 0) if 𝑢(𝑛)→0 as 𝑛→±∞. In addition, if 𝑢(𝑛)≢0, then 𝑢(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:𝑢̈(𝑡)+∇[−𝐾(𝑡,𝑢(𝑡))+𝑊(𝑡,𝑢(𝑡))]=0,𝑡∈ℝ.(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 Poincaré [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 𝑥(𝑛+1) is a function of the 𝑛th generation 𝑥(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 𝑊 satisfy the following conditions. (K1)There are two positive constants 𝑏₁ and 𝑏₂ such that 𝑏₁||𝑥|^𝑝≤𝐾(𝑛,𝑥)≤𝑏₂||𝑥|^𝑝,∀(𝑛,𝑥)∈ℤ×ℝ^𝑁.(1.3)(K2)There is a positive constant 𝑏₃ such that 𝑏₃|𝑥|^𝑝≤(∇𝐾(𝑛,𝑥),𝑥)≤||∇𝐾(𝑛,𝑥)|||𝑥|≤𝑝𝐾(𝑛,𝑥),∀(𝑛,𝑥)∈ℤ×ℝ^𝑁.(1.4)(W1)𝑊(𝑛,0)=0, ∇𝑊(𝑛,𝑥)=𝑜(|𝑥|^𝑝−1) as |𝑥|→0 uniformly for 𝑛∈ℤ.(W2)There exists a constant 𝑅>0 such that ||∇𝑊(𝑛,𝑥)|||𝑥|^𝑝−1≤𝑅,∀𝑛∈ℤ,𝑥∈ℝ^𝑁.(1.5)(W3)There exists a function 𝑉_∞∈𝑙^∞(ℤ,ℝ⁺) such that lim_|𝑥|→∞||∇𝑊(𝑛,𝑥)−𝑉_∞(𝑛)|𝑥|^𝑝−2𝑥|||𝑥|^𝑝−1inf_ℤ=0uniformlyfor𝑛∈ℤ,𝑉_∞(𝑛)>max1,𝑝𝑏₂.(1.6)(W4)𝑊(𝑛,𝑥)=(∇𝑊(𝑛,𝑥),𝑥)−𝑝𝑊(𝑛,𝑥), lim_|𝑥|→∞𝑊(𝑛,𝑥)=+∞uniformlyfor𝑛∈ℤ,(1.7) and for any fixed 0<𝑐₁<𝑐₂<+∞, inf_{n∈ℤ,𝑐1≤|𝑥|≤𝑐2}𝑊(𝑛,𝑥)|𝑥|^𝑝>0.(1.8)Then problem (1.1) has at least one nontrivial homoclinic solution.

Remark 1.2. The function 𝑊(𝑛,𝑥) in this paper is asymptotically 𝑝-linear at infinity. The behavior of the gradient of 𝑊(𝑛,𝑥) at infinity is like that of a function 𝑉_∞(𝑛)|𝑥|^𝑝−2𝑥, where 𝑉_∞(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 𝑊(𝑛,𝑥) at infinity cannot be found in the literature. From this point, our result is new.

2. Preliminaries

Let 𝐸𝑆=={𝑢𝑢(∈𝑛)𝑆}∶_𝑛∈ℤ∶_𝑛∈ℤ𝑢(𝑛)∈ℝ^𝑁,𝑛∈ℤ,||Δ𝑢(𝑛−1)||^𝑝+||𝑢(𝑛)||^𝑝<+∞,(2.1) and for 𝑢∈𝐸, let ‖𝑢‖=_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+||𝑢(𝑛)||^𝑝<+∞^1/𝑝.(2.2) Then 𝐸 is a uniform convex Banach space with this norm. As usual, for 1≤𝑝<+∞, let 𝑙^𝑝ℤ,ℝ^𝑁=𝑢∈𝑆∶_𝑛∈ℤ||𝑢(𝑛)||^𝑝<+∞,𝑙^∞ℤ,ℝ^𝑁=𝑢∈𝑆∶sup_𝑛∈ℤ||𝑢(𝑛)||<+∞,(2.3) and their norms are given by ‖𝑢‖_𝑙^𝑝=_𝑛∈ℤ||𝑢(𝑛)||^𝑝^1/𝑝‖𝑢‖_∞,∀𝑢∈𝑙^𝑝ℤ,ℝ^𝑁,=sup||𝑢(𝑛)||∶𝑛∈ℤ,∀𝑢∈𝑙^∞ℤ,ℝ^𝑁,(2.4) respectively.

For any 𝑢∈𝐸, let𝜑(𝑢)=1𝑝_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝−_𝑛∈ℤ[−𝐾(𝑛,𝑢(𝑛))+𝑊(𝑛,𝑢(𝑛))].(2.5)

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

Lemma 2.1 (see [29]). Let {𝑓_𝑘(𝑡)} and {𝑔_𝑘(𝑡)} be two sequences of measurable functions on a measurable set 𝐴, and let ||𝑓_𝑘(𝑡)||≤𝑔_𝑘(𝑡),∀𝑎.𝑒.𝑡∈𝐴.(2.6) If lim_𝑘→∞𝑓_𝑘(𝑡)lim=_𝑘→∞𝑓(𝑡),lim_𝑘→∞𝑔_𝑘(𝑡)=𝑔(𝑡),∀𝑎.𝑒.𝑡∈𝐴,_𝐴𝑔_𝑘(𝑡)𝑑𝑡=_𝐴𝑔(𝑡)𝑑𝑡<+∞,(2.7) then lim_𝑘→∞_𝐴𝑓_𝑘(𝑡)𝑑𝑡=_𝐴𝑓(𝑡)𝑑𝑡.(2.8)

Lemma 2.2. For 𝑢∈𝐸, ‖𝑢‖_∞≤‖𝑢‖_𝑙^𝑝≤2‖𝑢‖.(2.9)

Proof. Since 𝑢∈𝐸, it follows that lim_|𝑛|→∞|𝑢(𝑛)|=0. Hence, there exists 𝑛^∗∈ℤ such that ‖𝑢‖_∞=||𝑢𝑛^∗||=max_𝑛∈ℤ||𝑢(𝑛)||.(2.10) Hence, we have ‖𝑢‖_∞≤_𝑛∈ℤ||𝑢(𝑛)||^𝑝^1/𝑝=‖𝑢‖_𝑙^𝑝=_𝑛∈ℤ||𝑢(𝑛)−𝑢(𝑛−1)+𝑢(𝑛−1)||^𝑝^1/𝑝≤_𝑛∈ℤ||𝑢(𝑛)−𝑢(𝑛−1)||+||𝑢(𝑛−1)||^𝑝^1/𝑝≤2^𝑝_𝑛_∈ℤ||𝑢(𝑛)−𝑢(𝑛−1)||^𝑝+||𝑢(𝑛−1)||^𝑝^1/𝑝=2_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+||𝑢(𝑛−1)||^𝑝^1/𝑝=2_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+||𝑢(𝑛)||^𝑝^1/𝑝=2‖𝑢‖.(2.11)

Lemma 2.3. Suppose that (K1), (K2), and (W2) hold. If 𝑢_𝑘→𝑢 in 𝐸, then ∇𝐾(𝑛,𝑢_𝑘)→∇𝐾(𝑛,𝑢) and ∇𝑊(𝑛,𝑢_𝑘)→∇𝑊(𝑛,𝑢) in 𝑙^𝑝(ℝ,ℝ^𝑁), where 𝑝^>1 satisfies 1/𝑝+1/𝑝^=1.

Proof. From (K1) and (K2), we have ||∇𝐾(𝑛,𝑥)||≤𝑝𝑏₂|𝑥|^𝑝−1,∀(𝑛,𝑥)∈ℤ×ℝ^𝑁.(2.12) Hence, from (2.12), we have ||∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛))||^𝑝≤𝑝𝑏₂||𝑢^𝑘(𝑛)||^𝑝−1+||𝑢(𝑛)||^𝑝−1^𝑝≤𝑝𝑏₂2^𝑝−1||𝑢_𝑘(𝑛)−𝑢(𝑛)||^𝑝−1+𝑝𝑏₂1+2^𝑝−1||𝑢(𝑛)||^𝑝−1^𝑝≤2^𝑝𝑝𝑝𝑏₂^𝑝||𝑢_𝑘(𝑛)−𝑢(𝑛)||^𝑝+2^𝑝𝑝𝑏₂^𝑝1+2^𝑝−1^𝑝||𝑢(𝑛)||^𝑝∶=𝑔_𝑘(𝑛).(2.13) Moreover, since 𝑢_𝑘→𝑢 in 𝑙^𝑝(ℤ,ℝ^𝑁) and 𝑢_𝑘(𝑛)→𝑢(𝑛) for almost every 𝑛∈ℤ, hence, lim_𝑘→∞lim_𝑘→∞𝑔_𝑘(𝑛)=2^𝑝𝑝𝑏₂^𝑝1+2^𝑝−1^𝑝||𝑢(𝑛)||^𝑝∶=𝑔(𝑛),∀a.e.𝑛∈ℤ,_𝑛_∈ℤ𝑔_𝑘(𝑛)=lim_{𝑘→∞𝑛}_∈ℤ2^𝑝𝑝𝑝𝑏₂^𝑝||𝑢_𝑘(𝑛)−𝑢(𝑛)||^𝑝+2^𝑝𝑝𝑏₂^𝑝1+2^𝑝−1^𝑝||𝑢(𝑛)||^𝑝=2_𝑝𝑝𝑝𝑏₂^𝑝lim_𝑘→∞_𝑛∈ℤ||𝑢_𝑘(𝑛)−𝑢(𝑛)||^𝑝+2^𝑝𝑝𝑏₂^𝑝1+2^𝑝−1^𝑝_𝑛∈ℤ||𝑢(𝑛)||^𝑝=2^𝑝𝑝𝑏₂^𝑝1+2^𝑝−1^𝑝_𝑛_∈ℤ||𝑢(𝑛)||^𝑝=_𝑛∈ℤ𝑔(𝑛)<+∞.(2.14) It follows from Lemma  2.1, (2.13), and the previous equations that lim_{𝑘→∞𝑛}_∈ℤ||∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛))||^𝑝=0.(2.15) This shows that ∇𝐾(𝑛,𝑢_𝑘)→∇𝐾(𝑛,𝑢) in 𝑙^𝑝(ℤ,ℝ^𝑁). By a similar proof, we can prove that ∇𝑊(𝑛,𝑢_𝑘)→∇𝑊(𝑛,𝑢) in 𝑙^𝑝(ℤ,ℝ^𝑁). The proof is complete.

Lemma 2.4. Under the conditions of Theorem 1.1, one has ⟨𝜑′(𝑢),𝑣⟩=_𝑛∈ℤ+||Δ(∇𝑢𝐾(𝑛(−𝑛,1𝑢)||(𝑛)^𝑝−2)(−Δ𝑢∇(𝑊𝑛−(𝑛1,)𝑢,(Δ𝑛𝑣))(,𝑛𝑣−(𝑛1))))(2.16) for 𝑢,𝑣∈𝐸, which yields that ⟨𝜑′(𝑢),𝑢⟩=_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+(∇𝐾(𝑛,𝑢(𝑛)),𝑢(𝑛))−(∇𝑊(𝑛,𝑢(𝑛)),𝑢(𝑛)).(2.17) Moreover, 𝜑 is continuously Fréchet-differential defined on 𝐸; that is, 𝜑∈𝐶¹(𝐸,ℝ) and any critical point 𝑢 of 𝜑 on 𝐸 is classical solution of (1.1) with 𝑢(±∞)=0.

Proof. Firstly, we show that 𝜑∶𝐸→ℝ. Let 𝑢∈𝐸, by (2.9) and (K1), we have _𝑛_∈ℤ𝐾(𝑛,𝑢(𝑛))≤_𝑛∈ℤ𝑏₂||𝑢(𝑛)||^𝑝≤𝑏₂2^𝑝‖𝑢‖^𝑝.(2.18) By (W2), we get ||𝑊(𝑛,𝑥)||=||||¹⁰(∇𝑊(𝑛,𝑠𝑥),𝑥)𝑑𝑠||||≤𝑅|𝑥|^𝑝,∀(𝑛,𝑥)∈ℤ×ℝ^𝑁.(2.19) Hence, from (2.9) and (2.19), we have |||||_𝑛∈ℤ𝑊(𝑛,𝑢(𝑛))|||||≤_𝑛∈ℤ||𝑊(𝑛,𝑢(𝑛))||≤_𝑛∈ℤ𝑅||𝑢(𝑛)||^𝑝≤𝑅2^𝑝‖𝑢‖^𝑝.(2.20) It follows from (2.5), (2.18), and (2.20) that 𝜑∶𝐸→ℝ. Next we prove that 𝜑∈𝐶¹(𝐸,ℝ). Rewrite 𝜑 as follows: 𝜑(𝑢)=𝜑₁(𝑢)+𝜑₂(𝑢)−𝜑₃(𝑢),(2.21) where 𝜑₁(𝑢)∶=1𝑝_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝,𝜑₂(𝑢)∶=_𝑛∈ℤ𝐾(𝑛,𝑢(𝑛)),𝜑₃(𝑢)∶=_𝑛∈ℤ𝑊(𝑛,𝑢(𝑛)).(2.22) It is easy to check that 𝜑₁∈𝐶¹(𝐸,ℝ) and 𝜑^1(𝑢),𝑣=_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝−2(Δ𝑢(𝑛−1),Δ𝑣(𝑛−1)),∀𝑢,𝑣∈𝐸.(2.23) Next, we prove that 𝜑_𝑖∈𝐶¹(𝐸,ℝ),𝑖=2,3, and 𝜑^2(𝑢),𝑣=_𝑛∈ℤ(∇𝐾(𝑛,𝑢(𝑛)),𝑣(𝑛)),∀𝑢,𝑣∈𝐸,(2.24)𝜑^3(𝑢),𝑣=_𝑛∈ℤ(∇𝑊(𝑛,𝑢(𝑛)),𝑣(𝑛)),∀𝑢,𝑣∈𝐸.(2.25) For any 𝑢,𝑣∈𝐸 and for any function 𝜃∶ℝ→(0,1), by (K2), we have _𝑛_∈ℤmax_[]ℎ∈0,1||(∇𝐾(𝑛,𝑢(𝑛)+𝜃(𝑡)ℎ𝑣(𝑛)),𝑣(𝑛))||≤𝑝𝑏_2𝑛_∈ℤ≤2_𝑝−1max_[]ℎ∈0,1||𝑢(𝑛)+𝜃(𝑡)ℎ𝑣(𝑛)||^𝑝−1||𝑣(𝑛)||𝑝𝑏_2𝑛_∈ℤ||𝑢(𝑛)||^𝑝−1≤2_𝑝−1+||𝑣(𝑛)||^𝑝−1||𝑣(𝑛)||<+∞.𝑝𝑏₂‖𝑢‖^{𝑙𝑝−1𝑝}‖𝑣‖_𝑙^𝑝+‖𝑣‖^𝑝𝑙_𝑝(2.26) Then by the previous equations and Lebesgue's dominated convergence theorem, we have 𝜑^2(𝑢),𝑣=lim_ℎ→0+𝜑₂(𝑢+ℎ𝑣)−𝜑₂(𝑢)=lim_ℎ→0+ℎ1ℎ_𝑛∈ℤ=lim_ℎ→0+[𝐾(𝑛,𝑢(𝑛)+ℎ𝑣(𝑛))−𝐾(𝑛,𝑢(𝑛))]_𝑛_∈ℤ=(∇𝐾(𝑡,𝑢(𝑛)+𝜃(𝑡)ℎ𝑣(𝑛)),𝑣(𝑛))_𝑛∈ℤ(∇𝐾(𝑛,𝑢(𝑛)),𝑣(𝑛)),∀𝑢,𝑣∈𝐸.(2.27) Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that 𝜑_𝑖∈𝐶¹(𝐸,ℝ),𝑖=2,3. Let 𝑢_𝑘→𝑢 in 𝐸; then by Lemma 2.3, we have ||𝜑^2=𝑢|||||_𝑘−𝜑^2(𝑢),𝑣_𝑛∈ℤ||≤∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑣(𝑛)_𝑛∈ℤ|||||≤‖𝑣‖||∇𝐾_𝑛∈ℤ𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛))‖𝑣(𝑛)||||∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛))||^𝑝^1/𝑝⟶0,𝑘⟶∞,∀𝑣∈𝐸.(2.28) This shows that 𝜑₂∈𝐶¹(𝐸,ℝ). Similarly, we can prove that 𝜑₃∈𝐶¹(𝐸,ℝ). Furthermore, by a standard argument, it is easy to show that the critical points of 𝜑 in 𝐸 are classical solutions of (1.1) with 𝑢(±∞)=0. The proof is complete.

Lemma 2.5 (see [30]). Let 𝐸 be a real Banach space with its dual space 𝐸^∗ and suppose that 𝜑∈𝐶¹(𝐸,ℝ) satisfies max{𝜑(0),𝜑(𝑒)}≤𝜂₀<𝜂≤inf_{‖𝑢‖=𝜌}𝜑(𝑢),(2.29) for some 𝜂₀<𝜂, 𝜌>0, and 𝑒∈𝐸 with ‖𝑒‖>𝜌. Let 𝑐≥𝜂 be characterized by 𝑐=inf_Υ∈Γmax_0≤𝜏≤1𝜑(Υ(𝜏)),(2.30) where Γ={Υ∈𝐶([0,1],𝐸)∶Υ(0)=0,Υ(1)=𝑒} is the set of continuous paths joining 0 to 𝑒; then there exists {𝑢_𝑘}_𝑘∈ℕ⊂𝐸 such that 𝜑𝑢_𝑘⟶𝑐,1+‖‖𝑢_𝑘‖‖‖‖𝜑′(𝑢_𝑘)‖‖_𝐸∗⟶0as𝑘⟶∞.(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 ∇𝑊(𝑛,𝑥)≤𝐶¹2^𝑝|𝑥|^𝑝−1,∀𝑛∈ℤ,|𝑥|≤𝜌₀,(3.1) where 𝐶₁=min{1/𝑝,𝑏₁}. From (3.1), we have 𝑊(𝑛,𝑥)=≤¹⁰(∇𝑊(𝑛,𝑠𝑥),𝑥)𝑑𝑠₁₀𝐶₁2^𝑝|𝑥|^𝑝𝑠^𝑝−1𝑑𝑠=𝐶¹𝑝2^𝑝|𝑥|^𝑝,∀𝑛∈ℤ,|𝑥|≤𝜌₀.(3.2) Let 𝜌=𝜌₀/2 and 𝑆={𝑢∈𝐸∣‖𝑢‖=𝜌}; then from (2.9), we obtain ‖𝑢‖_∞≤𝜌₀,‖𝑢‖_𝑙^𝑝≤2𝜌,∀𝑢∈𝑆,(3.3) which together with (2.9), (3.2), and (K1) implies that 𝜑(𝑢)=1𝑝_𝑛∈ℤ≥1||Δ𝑢(𝑛−1)||^𝑝−_𝑛∈ℤ[−𝐾(𝑛,𝑢(𝑛))+𝑊(𝑛,𝑢(𝑛))]𝑝_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+𝑏_1𝑛_∈ℤ||𝑢(𝑛)||^𝑝−_𝑛∈ℤ𝐶₁𝑝2^𝑝||𝑢(𝑛)||^𝑝≥min1𝑝,𝑏¹‖𝑢‖^𝑝−𝐶¹𝑝2^𝑝‖𝑢‖^𝑝𝑙_𝑝≥𝐶₁‖𝑢‖^𝑝−𝐶¹𝑝‖𝑢‖^𝑝=(𝑝−1)𝐶¹𝑝‖𝑢‖^𝑝=𝛼₁>0,𝑢∈𝑆.(3.4)Step 2. From (K1), we have 𝜑(𝑢)=1𝑝_𝑛∈ℤ≤1||Δ𝑢(𝑛−1)||^𝑝−_𝑛∈ℤ[−𝐾(𝑛,𝑢(𝑛))+𝑊(𝑛,𝑢(𝑛))]𝑝_𝑛∈ℤ||Δ𝑢(𝑛−1)||^𝑝+𝑏_2𝑛_∈ℤ≤max1||𝑢(𝑛)||^𝑝−_𝑛∈ℤ𝑊(𝑛,𝑢(𝑛))𝑝,𝑏²‖𝑢‖^𝑝≡𝐶₂−_𝑛∈ℤ𝑊(𝑛,𝑢(𝑛))‖𝑢‖^𝑝−_𝑛∈ℤ𝑊(𝑛,𝑢(𝑛)).(3.5) By (W2) and (W3), we get lim_|𝑥|→∞𝑝𝑊(𝑛,𝑥)|𝑥|^𝑝=𝑉_∞(𝑛)uniformlyfor𝑛∈ℤ.(3.6) Let 𝑊(𝑛,𝑥)=𝑝𝑊(𝑛,𝑥)−𝑉_∞(𝑛)|𝑥|^𝑝; it follows from (W2), (W3), (2.19), and (3.6) that 𝑊(𝑛,𝑥)≤𝑝𝑅+sup_ℤ𝑉_∞(𝑛)|𝑥|^𝑝,∀𝑥∈ℝ^𝑁,lim_|𝑥|→∞𝑊(𝑛,𝑥)|𝑥|^𝑝=0.(3.7) Define 𝐸₁∶={𝑢(𝑛)=𝑥𝑒^−|𝑛|∶𝑥∈ℝ^𝑁,𝑛∈ℤ}⊂𝐸 with inf_ℤ𝑉_∞(𝑛)>max1,𝑝𝑏₂1+||1−𝑒^{|𝑛|−|𝑛−1|}||^𝑝.(3.8) By an easy calculation, we have ‖𝑢‖^𝑝=1+||1−𝑒^{|𝑛|−|𝑛−1|}||^𝑝‖𝑢‖^𝑝𝑙𝑝.(3.9) In what follows, we prove that for some 𝑢∈𝐸₁ with ‖𝑢‖=1, 𝜑(𝑠𝑢)→−∞ as 𝑠→∞. Otherwise, there exist a sequence {𝑠_𝑘} with 𝑠_𝑘→∞ as 𝑘→∞ and a positive constant 𝐶₃ such that 𝜑(𝑠_𝑘𝑢)≥−𝐶₃ for all 𝑘. From (3.5), we obtain −𝐶³𝑠^𝑝𝑘≤𝜑𝑠^𝑘𝑢𝑠^𝑝𝑘≤𝐶₂−1𝑝_𝑛∈ℤ𝑊𝑛,𝑠_𝑘𝑢(𝑛)𝑠^𝑝𝑘−1𝑝_𝑛∈ℤ𝑉_∞(𝑛)||𝑢(𝑛)||^𝑝≤𝐶₂−1𝑝_𝑛∈ℤ𝑊𝑛,𝑠_𝑘𝑢(𝑛)𝑠^𝑝𝑘−1𝑝inf^ℤ𝑉_∞(𝑛)‖𝑢‖^𝑝𝑙_𝑝.(3.10) It follows from (3.7) that 𝑊𝑛,𝑠_𝑘𝑢(𝑛)𝑠^𝑝𝑘≤𝑝𝑅+sup_ℤ𝑉_∞(𝑛)||𝑢(𝑛)||^𝑝,𝑊𝑛,𝑠_𝑘𝑢(𝑛)||𝑠_𝑘||^𝑝⟶0as𝑘⟶∞.(3.11) Hence, from Lebesgue's dominated theorem and (3.11), we have _𝑛_∈ℤ𝑊𝑛,𝑠_𝑛𝑢(𝑛)||𝑠_𝑘||^𝑝⟶0as𝑘⟶∞.(3.12) It follows from (3.8), (3.9), (3.10), and (3.12) that 0⟵−𝐶³𝑠^𝑝𝑘≤𝐶₂−1𝑝1+||1−𝑒^{|𝑛|−|𝑛−1|}||^𝑝inf^ℤ𝑉_∞(𝑛)<0as𝑘⟶∞,(3.13) which is a contradiction. Hence, there exists 𝑒∈𝐸 with ‖𝑒‖>𝜌 such that 𝜑(𝑒)≤0.Step 3. From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence {𝑢_𝑘}_𝑘∈ℕ⊂𝐸 such that 𝜑𝑢_𝑘⟶𝑐,1+‖‖𝑢_𝑘‖‖‖‖𝜑^𝑢_𝑘‖‖_𝐸^∗⟶0as𝑘⟶∞,(3.14) where 𝐸^∗ is the dual space of 𝐸. In the following, we will prove that {𝑢_𝑘}_𝑘∈ℕ is bounded in 𝐸. Otherwise, assume that ‖𝑢_𝑘‖→∞ as 𝑘→∞. Let 𝑧_𝑘=𝑢_𝑘/‖𝑢_𝑘‖; we have ‖𝑧_𝑘‖=1. It follows from (2.5), (2.16), (3.14), and (K2) that 𝐶₄≥=𝑝𝜑_𝑛∈ℤ𝑢_𝑘−𝜑′𝑢_𝑘,𝑢_𝑘+∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)−𝑝𝑊𝑛,𝑢_𝑘(𝑛)_𝑛∈ℤ≥𝑝𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)_𝑛∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)−𝑝𝑊𝑡,𝑢_𝑘(𝑛)∶=_𝑛∈ℤ𝑊𝑛,𝑢_𝑘(𝑛).(3.15) Set Ω_𝑘(𝛼,𝛽)={𝑛∈ℤ∶𝛼≤|𝑢_𝑘(𝑛)|≤𝛽} for 0<𝛼<𝛽. Then from (3.15), we have 𝐶₄≥_{𝑛∈Ω𝑘(0,𝛼)}𝑊𝑛,𝑢_𝑘(𝑛)+_{𝑛∈Ω𝑘(𝛼,𝛽)}𝑊𝑛,𝑢_𝑘(𝑛)+_{𝑛∈Ω𝑘(𝛽,+∞)}𝑊𝑛,𝑢_𝑘(𝑛).(3.16) From (K1), (K2), and (3.14), we get 𝑜(1)==𝜑_𝑛∈ℤ′𝑢_𝑘,𝑢_𝑘≥||Δ𝑢_𝑘(𝑛−1)||^𝑝+∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)_𝑛∈ℤ≥min||Δ1𝑢,_𝑘𝑏(₃𝑛−1)||^𝑝+𝑏₃||𝑢_𝑘(𝑛)||^𝑝−∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)∶=𝐶₅‖‖𝑢_𝑘‖‖^𝑝−_𝑛∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)=‖‖𝑢‖‖𝑢_𝑘‖‖^𝑝−_𝑛∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)_𝑘‖‖^𝑝𝐶⁵−_𝑛∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)‖‖𝑢_𝑘‖‖^𝑝,(3.17) which implies that limsup_{𝑘→∞𝑛}_∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)||𝑢_𝑘(𝑛)||^𝑝||𝑧_𝑘(𝑛)||^𝑝=limsup_{𝑘→∞𝑛}_∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)‖‖𝑢_𝑘‖‖^𝑝≥𝐶₅.(3.18) Let 0<𝜀<𝐶₅/3. From (W1), there exists 𝛼_𝜀>0 such that ||∇𝑊(𝑛,𝑥)||≤𝜀2^𝑝|𝑥|^𝑝−1for|𝑥|≤𝛼_𝜀uniformlyfor𝑛∈ℤ.(3.19) Since ‖𝑧_𝑘‖=1, it follows from (2.9) and (3.19) that _𝑛∈Ω_{𝑘(0,𝛼𝜀)}||∇𝑊𝑛,𝑢_𝑘(𝑛)||||𝑢_𝑘(𝑛)||^𝑝−1||𝑧_𝑘(𝑛)||^𝑝≤_{𝑛∈Ω𝑘(0,𝛼𝜀)}𝜀2^𝑝||𝑧_𝑘(𝑛)||^𝑝≤𝜀,∀𝑘∈ℕ.(3.20) For 𝑠>0, let ℎ(𝑠)∶=inf𝑊(𝑛,𝑥)∣𝑛∈ℤ,𝑥∈ℝ^𝑁with|𝑥|≥𝑠.(3.21) Thus, from (W4), we have ℎ(𝑠)→+∞ as 𝑠→+∞, which together with (3.16) implies that measΩ_𝑘(𝛽,+∞)≤𝐶⁶ℎ(𝛽)⟶0,as𝛽⟶+∞.(3.22) Hence, we can take 𝛽_𝜀 sufficiently large such that _{𝑛∈Ω𝑘}_{𝛽𝜀,+∞}||𝑧_𝑘(𝑛)||^𝑝<𝜀𝑅.(3.23) The previous inequality and (W2) imply that _{𝑛∈Ω𝑘}_{𝛽𝜀,+∞}||∇𝑊𝑛,𝑢_𝑘(𝑛)||||𝑢_𝑘(𝑛)||^𝑝−1||𝑧_𝑘(𝑛)||^𝑝≤𝑅_{𝑛∈Ω𝑘𝛽𝜀,+∞}||𝑧_𝑘(𝑛)||^𝑝<𝜀,∀𝑘∈ℕ.(3.24) Next, for the previous 0<𝛼_𝜀<𝛽_𝜀, let 𝑐_𝜀∶=inf𝑊(𝑛,𝑥)|𝑥|^𝑝∶𝑛∈ℤ,𝑥∈ℝ^𝑁with𝛼_𝜀≤|𝑥|≤𝛽_𝜀𝑑_𝜀,∶=max||∇𝑊(𝑛,𝑥)|||𝑥|^𝑝−1∶𝑛∈ℤ,𝑥∈ℝ^𝑁with𝛼_𝜀≤|𝑥|≤𝛽_𝜀.(3.25) From (W4), we have 𝑐_𝜀>0 and 𝑊𝑛,𝑢_𝑘(𝑛)≥𝑐_𝜀||𝑢_𝑘(𝑛)||^𝑝,∀𝑛∈Ω_𝑘𝛼_𝜀,𝛽_𝜀.(3.26) From (3.15) and (3.26), we get _𝑛∈Ω_{𝑘(𝛼𝜀,𝛽𝜀)}||𝑧_𝑘(𝑛)||^𝑝=1‖‖𝑢_𝑘‖‖^𝑝_{𝑛∈Ω𝑘(𝛼𝜀,𝛽𝜀)}||𝑢_𝑘(𝑛)||^𝑝≤1‖‖𝑢_𝑘‖‖^𝑝_{𝑛∈Ω𝑘𝛼𝜀,𝛽𝜀}1𝑐_𝜀≤𝐶𝑊𝑛,𝑢_𝑘(𝑛)₄𝑐_𝜀‖‖𝑢_𝑘‖‖^𝑝⟶0as𝑘⟶∞,(3.27) which implies that _𝑛∈Ω_{𝑘𝛼𝜀,𝛽𝜀}||∇𝑊𝑛,𝑢_𝑘(𝑛)||||𝑢_𝑘(𝑛)||^𝑝−1||𝑧_𝑘(𝑛)||^𝑝≤𝑑_{𝜀𝑛∈Ω}_{𝑘𝛼𝜀,𝛽𝜀}||𝑧_𝑘(𝑛)||^𝑝⟶0as𝑘⟶∞.(3.28) Therefore, there exists 𝑘₀>0 such that _𝑛∈Ω_{𝑘𝛼𝜀,𝛽𝜀}||∇𝑊𝑛,𝑢_𝑘(𝑛)||||𝑢_𝑘(𝑛)||^𝑝−1||𝑧_𝑘(𝑛)||^𝑝≤𝜀,∀𝑘≥𝑘₀.(3.29) It follows from (3.20), (3.24), and (3.29) that _𝑛_∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)||𝑢_𝑘(𝑛)||^𝑝||𝑧_𝑘(𝑛)||^𝑝≤_𝑛∈ℤ||∇𝑊𝑡,𝑢_𝑘(𝑛)||||𝑢_𝑘(𝑛)||^𝑝−1||𝑧_𝑘(𝑛)||^𝑝<3𝜀<𝐶₅,∀𝑘≥𝑘₀,(3.30) which implies that limsup_{𝑛→∞𝑛}_∈ℤ∇𝑊𝑛,𝑢_𝑘(𝑛),𝑢_𝑘(𝑛)||𝑢_𝑘(𝑛)||^𝑝||𝑧_𝑘(𝑛)||^𝑝<𝐶₅,(3.31) but this contradicts to (3.18). Hence, ‖𝑢_𝑘‖ is bounded in 𝐸.
Going to a subsequence if necessary, we may assume that there exists 𝑢∈𝐸 such that 𝑢_𝑘⇀𝑢 as 𝑘→∞. In order to prove our theorem, it is sufficient to show that 𝜑′(𝑢)=0. For any 𝑎∈ℤ with 𝑎>0, let 𝜒_𝑎(𝑡)=1 for 𝑡∈ℤ[−𝑎,𝑎] and let 𝜒_𝑎(𝑡)=0 for 𝑡∈ℤ(−∞,−𝑎)∪ℤ(𝑎,∞). Then from (2.16), we have 𝜑′=𝑢_𝑘−𝜑′(𝑢),𝜒_𝑎𝑢_𝑘−𝑢_{[]𝑛∈ℤ−𝑎,𝑎}−||Δ𝑢_𝑘(𝑛−1)||^𝑝−2Δ𝑢_𝑘(𝑛−1),Δ𝑢_𝑘(𝑛−1)−Δ𝑢(𝑛−1)_{[]𝑛∈ℤ−𝑎,𝑎}+||Δ𝑢(𝑛−1)||^𝑝−2Δ𝑢(𝑛−1),Δ𝑢_𝑘(𝑛−1)−Δ𝑢(𝑛−1)_{[]𝑛∈ℤ−𝑎,𝑎}−∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)_{[]𝑛∈ℤ−𝑎,𝑎}≥‖‖Δ𝑢_𝑘∇𝑊𝑛,𝑢_𝑘(𝑛)−∇𝑊(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)−‖‖^𝑝𝑙_{𝑝ℤ[]−𝑎,𝑎}+‖Δ𝑢‖^𝑝𝑙_{𝑝ℤ[]−𝑎,𝑎}−_{[]𝑛∈ℤ−𝑎,𝑎}||Δ𝑢_𝑘(𝑛−1)||^𝑝−1||Δ𝑢(𝑛−1)_{[]𝑛∈ℤ−𝑎,𝑎}||+||Δ𝑢(𝑛−1)||^𝑝−1||Δ𝑢_𝑘(𝑛−1)_{[]𝑛∈ℤ−𝑎,𝑎}||−∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)_{[]𝑛∈ℤ−𝑎,𝑎}≥‖Δ𝑢_𝑘∇𝑊𝑛,𝑢_𝑘(𝑛)−∇𝑊(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)‖^𝑝𝑙_{𝑝ℤ[]−𝑎,𝑎}+‖Δ𝑢‖^𝑝𝑙_{𝑝ℤ[−𝑎,𝑎]}−‖Δ𝑢‖_𝑙^𝑝_{ℤ[−𝑎,𝑎]}‖Δ𝑢_𝑘‖^𝑙𝑝−1_{𝑝ℤ[−𝑎,𝑎]}−‖Δ𝑢_𝑘‖_𝑙^𝑝_{ℤ[−𝑎,𝑎]}‖Δ𝑢‖^𝑙𝑝−1_{𝑝ℤ[−𝑎,𝑎]}+−_{𝑛∈ℤ[−𝑎,𝑎]}∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)_{[]𝑛∈ℤ−𝑎,𝑎}=‖‖Δ𝑢_𝑘∇𝑊𝑛,𝑢_𝑘(𝑛)−∇𝑊(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)+‖‖^𝑙𝑝−1_{𝑝ℤ[]−𝑎,𝑎[]𝑛∈ℤ−𝑎,𝑎}−‖Δ𝑢‖^𝑙𝑝−1_{𝑝ℤ[]−𝑎,𝑎}‖‖Δ𝑢^𝑘‖‖_{𝑙𝑝ℤ[−𝑎,𝑎]}−‖Δ𝑢‖_𝑙^𝑝_{ℤ[−𝑎,𝑎]}−∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)_{[]𝑛∈ℤ−𝑎,𝑎}∇𝑊𝑛,𝑢_𝑘(𝑛)−∇𝑊(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛).(3.32) Since 𝜑′(𝑢_𝑘)→0 as 𝑘→+∞ and 𝑢_𝑘⇀𝑢 in 𝐸, it follows from (3.14) that 𝜑′𝑢_𝑘−𝜑′(𝑢),𝜒_𝑎𝑢_𝑘−𝑢⟶0as𝑘⟶∞,_{𝑛∈ℤ[−𝑎,𝑎]}∇𝐾𝑛,𝑢_𝑘(𝑛)−∇𝐾(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)⟶0as𝑘⟶∞,_{𝑛∈ℤ[−𝑎,𝑎]}∇𝑊𝑛,𝑢_𝑘(𝑛)−∇𝑊(𝑛,𝑢(𝑛)),𝑢_𝑘(𝑛)−𝑢(𝑛)⟶0as𝑘⟶∞.(3.33) It follows from (3.32) and (3.33) that ‖Δ𝑢_𝑘‖_𝑙^𝑝_{ℤ[−𝑎,𝑎]}→‖Δ𝑢‖_𝑙^𝑝_{ℤ[−𝑎,𝑎]} as 𝑘→+∞.
For any 𝑤∈𝐶^∞0(ℝ,ℝ^𝑁), and assume that for some 𝐴∈ℤ with 𝐴>0, supp(𝑤)⊂ℤ[−𝐴,𝐴]. Since |||lim||Δ_𝑘→∞𝑢Δ_𝑘𝑢_𝑘(𝑛−1)=Δ𝑢(𝑛−1),∀a.e.𝑛∈ℤ,≤𝑝(𝑛−−11)||^𝑝−2Δ𝑢_𝑘(𝑛−1),Δ𝑤(𝑛−1)|||𝑝||Δ𝑢^𝑘(𝑛−1)||^𝑝+1𝑝||Δ𝑤(𝑛−1)||^𝑝lim_𝑘→∞,∀𝑛∈ℤ,𝑘=1,2,…,_[]𝑛_{∈ℤ−𝐴,𝐴}𝑝−𝑝1||Δ𝑢^𝑘(𝑛−1)||^𝑝+1=𝑝−1𝑝||Δ𝑤(𝑛−1)||^𝑝𝑝lim^𝑘→∞‖‖Δ𝑢_𝑘‖‖^𝑝𝑙_{𝑝ℤ[]−𝐴,𝐴}+1𝑝‖Δ𝑤‖^{𝑝𝑙𝑝ℤ[]−𝐴,𝐴}=𝑝−1𝑝‖Δ𝑢‖^{𝑝𝑙𝑝ℤ[]−𝐴,𝐴}+1𝑝‖Δ𝑤‖^{𝑝𝑙𝑝ℤ[]−𝐴,𝐴}=_{[]𝑛∈ℤ−𝐴,𝐴}𝑝−𝑝1||Δ𝑢(𝑛−1)||^𝑝+1𝑝||Δ𝑤(𝑛−1)||^𝑝<+∞,(3.34) then, we have _𝑛∈ℤ_{[−𝐴,𝐴]}||Δ𝑢^𝑘⟶(𝑛−1)||^𝑝−2Δ𝑢_𝑘(𝑛−1),Δ𝑤(𝑛−1)_{𝑛∈ℤ[−𝐴,𝐴]}||Δ𝑢(𝑛−1)||^𝑝−2Δ𝑢(𝑛−1),Δ𝑤(𝑛−1)(3.35) as 𝑘→∞. Noting that _[]𝑛_{∈ℤ−𝐴,𝐴}∇𝐾𝑛,𝑢_𝑘(𝑛),𝑤(𝑛)⟶_{𝑛∈ℤ[−𝐴,𝐴]}(∇𝐾(𝑛,𝑢(𝑛)),𝑤(𝑛))as𝑘⟶∞,_{𝑛∈ℤ[−𝐴,𝐴]}∇𝑊𝑛,𝑢_𝑘(𝑛),𝑤(𝑛)⟶_{𝑛∈ℤ[−𝐴,𝐴]}(∇𝑊(𝑛,𝑢(𝑛)),𝑤(𝑛))as𝑘⟶∞.(3.36) Hence, we have ⟨𝜑′(𝑢),𝑤⟩=lim_𝑘→∞𝜑′𝑢_𝑘,𝑤=0,(3.37) which implies that 𝜑′(𝑢)=0; that is, 𝑢 is a critical point of 𝜑. From (K1) and (W1), we know that 𝑢≠0. In fact, if 𝑢=0, we have from (2.5), (K1), and (W1) that 𝜑(𝑢)=0. On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that 𝜑(𝑢)=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 𝐾(𝑛,𝑥)=1+1|𝑥|^3/2+1|𝑥|^3/2,𝑊(𝑛,𝑥)=𝑎(𝑛)|𝑥|^3/21−(ln(𝑒+1|𝑥|))^1/2,(4.1) where 𝑎∈𝑙^∞(ℤ,ℝ⁺) with inf_ℤ𝑎(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 ∇𝑊(𝑛,𝑥)=32𝑎(𝑛)|𝑥|^−1/2𝑥1−1(ln(𝑒+|𝑥|))^1/2+𝑎(𝑛)|𝑥|^1/2𝑥2(𝑒+|𝑥|)(ln(𝑒+|𝑥|))^3/2(∇𝑊(𝑛,𝑥),𝑥)−3,2𝑊(𝑛,𝑥)=2(𝑒+|𝑥𝑎|)(𝑛(l)n|𝑥(|𝑒+^5/2|𝑥|))^3/2.(4.2) Then it is easy to check that 𝑊 satisfies (W1)–(W4). Hence, 𝐾(𝑛,𝑥) and 𝑊(𝑛,𝑥) 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.