[Retracted] Three Homoclinic Solutions for Second-Order 𝑝-Laplacian Differential System

Abstract

We consider second-order 𝑝-Laplacian differential system. By using three critical points theorem, we establish the new criterion to guarantee that this 𝑝-Laplacian differential system has at least three homoclinic solutions. An example is presented to illustrate the main result.

1. Introduction

Let us consider the following second-order 𝑝-Laplacian differential system: 𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)^′−𝑠(𝑡)Φ_𝑝(𝑢(𝑡((𝑡))+𝜆𝑓𝑡,𝑢))=0, (𝑃) whereΦ_𝑝(𝑥)∶=|𝑥|^𝑝−2𝑥, 𝑝>1, 𝜌,𝑠∈𝐿^∞ withessinf𝜌>0andessinf𝑠>0,𝑓∶𝑅×𝑅^𝑛→𝑅^𝑛is continuous,𝑡∈𝑅, and 𝜆∈[0,+∞). As usual, we say that a solution𝑢(𝑡)of (𝑃) is nontrivial homoclinic (to 0) if𝑢(𝑡)≠0, 𝑢(𝑡)→0 and𝑢̇(𝑡)→0as𝑡→±∞.

In the past two decades, many authors have studied homoclinic orbits for the second-order Hamiltonian systems 𝑞̈(𝑡)+∇𝑉𝑡,𝑞(𝑡)=𝑓(𝑡), (1) and the existence and multiplicity of homoclinic solutions for (1) have been extensively investigated via critical point theory (see [1–15]). For instance, Yang et al. [5] have shown the existence of infinitely many homoclinic solutions for (1) by using fountain theorem.

Theorem A (see [5]). Assume that𝑓and𝑉satisfy the following conditions: (H1)𝑓(𝑡)=0and∇𝑉(𝑡,𝑞(𝑡))=−𝐿(𝑡)𝑞(𝑡)+∇𝑊(𝑡,𝑞(𝑡));(H2)𝐿∈𝐶(𝑅,𝑅^𝑛×𝑛)is a symmetric and positive definite matrix for all𝑡∈𝑅and there is a continuous function𝛼∶𝑅→𝑅such that𝛼(𝑡)>0for all𝑡∈𝑅and (𝐿(𝑡)𝑢,𝑢)≥𝛼(𝑡)|𝑢|²,𝛼(𝑡)⟶+∞ (2) as|𝑡|→∞;(H3)consider the following 𝑊(𝑡,𝑢)=𝑚(𝑡)|𝑢|^𝛾+𝑑|𝑢|^𝑝, (3) where𝑚∶𝑅→𝑅⁺is a positive continuous function such that𝑚∈𝐿^2/(2−𝛾)(𝑅,𝑅⁺)and1<𝛾<2, 𝑑≥0, and 𝑝>2are constants.
Then (1) possesses infinitely many homoclinic solutions.
Moreover, Tang and Xiao [10] prove the existence of homoclinic solution of (1) as a limit of the2𝑘𝑇-periodic solutions of the following extension of system (1): 𝑞̈(𝑡)=−∇𝑉𝑡,𝑞(𝑡)+𝑓_𝑘(𝑡), (4) and they established the following theorem.

Theorem B (see [10]). Assume that𝑓and𝑉satisfy the following conditions: (H4)𝑉,𝑓(𝑡)≠0and 𝑉(𝑡,𝑥)=−𝐾(𝑡,𝑥)+𝑊(𝑡,𝑥), where 𝑉∈𝐶¹(𝑅×𝑅^𝑛,𝑅) is T-periodic with respect to 𝑡, and 𝑇>0; (H5)∇𝑊(𝑡,𝑥)=𝑜(|𝑥|), as|𝑥|→0 uniformly with respect to 𝑡; (H6) there is a constant𝜇>2such that 0<𝜇𝑊(𝑡,𝑥)≤(𝑥,∇𝑊(𝑡,𝑥))∀(𝑡,𝑥)∈𝑅×𝑅^𝑛⧵{0}; (5)(H7)𝑓∶𝑅×𝑅^𝑛 is a continuous and bounded function;(H8) there exist constants𝑏>0 and 𝛾∈(1,2]such that 𝐾(𝑡,0)=0,𝐾(𝑡,𝑥)≥𝑏|𝑥|^𝛾for(𝑡,𝑥)∈[0,𝑇]×𝑅^𝑛; (6) (H9) there is a constant 𝜚∈[2,𝜇)such that (𝑥,∇𝐾(𝑡,𝑥))≤𝜚𝐾(𝑡,𝑥),for(𝑡,𝑥)∈[0,𝑇]×𝑅^𝑛; (7) (H10)consider the following _𝑅𝑓(𝑡)²𝛿𝑑𝑡<2min2,𝑏𝛿^𝛾−1−𝑀𝛿^𝜇−12. (8)
Then system (1) possesses a nontrivial homoclinic solution.

For 𝑝-Laplacian problem, Tian and Ge [16] obtained sufficient conditions that guarantee the existence of at least two positive solutions of 𝑝-Laplacian boundary value problem with impulsive effects. Two key conditions of the main results of [16] are listed as follows: (H11) there exist𝜇>𝑝, ℎ∈𝐶([𝑎,𝑏]×[0,+∞),[0,+∞)), 𝜂>0, 𝑟∈𝐶([𝑎,𝑏]×[0,+∞)), 𝑔∈𝐶([0,+∞),[0,+∞)), and ^𝑏_𝑎𝑟(𝑠)𝑑𝑠+𝜂>0, (9) such that 𝑓(𝑡,𝑥)=𝑟(𝑡)Φ_𝜇(𝑥)(),𝐼+ℎ𝑡,𝑥_𝑖(𝑥)=𝜂Φ_𝜇(𝑥)+𝑔(𝑥); (10)(H12) there exist𝑐∈𝐿¹([𝑎,𝑏],[0,+∞)), 𝑑∈𝐶([𝑎,𝑏],[0,+∞)), 𝜉≥0, such that ℎ(𝑡,𝑥)≤𝑐(𝑡)+𝑑(𝑡)Φ_𝑝(𝑥). (11)

In [17], Ricceri established a three critical points theorem. After that, several authors used it to obtain some interesting results (see [18–22]).

Existence and multiplicity of solutions for 𝑝-Laplacian boundary value problem have been studied extensively in the literature (see [23–26]). However, to our best knowledge, the existence of at least three homoclinic solutions for 𝑝-Laplacian differential system has attracted less attention.

Motivated by the aforementioned facts, in this paper we are devoted to study the multiplicity homoclinic solutions of (𝑃) via three critical points theorem obtained by Ricceri [17].

In order to receive the homoclinic solution of (𝑃), similar to [10] we consider a sequence of system of differential equations as follows: 𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)^′−𝑠(𝑡)Φ_𝑝(𝑢(𝑡))+𝜆𝑓_(𝑘)(𝑡,𝑢(𝑡))=0, 𝑃_𝑘 where𝑓_(𝑘)∶𝑅×𝑅^𝑛→𝑅^𝑛is a 2𝑘𝑇-periodic extension of restriction of𝑓to the interval[−𝑘𝑇,𝑘𝑇), 𝑘∈𝑁. We will prove the existence of three homoclinic solutions of (𝑃) as the limit of the 2𝑘𝑇-periodic solutions of 𝑃_𝑘 as in [10]. However, many technical details in our paper are different from [10, 12].

2. Preliminaries

For each𝑘∈𝑁, let𝐸_(𝑘)=𝑊^1,𝑝_2𝑘𝑇(𝑅,𝑅^𝑛)denote the Sobolev space of 2𝑘𝑇-periodic functions on𝑅with values in𝑅^𝑛under the norm ‖𝑢‖∶=‖𝑢‖_𝐸(𝑘)=^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢^′(𝑡)^𝑝+𝑠(𝑡)|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝, (12) which is equivalent to the usual one. We define the norm in𝐶([−𝑘𝑇,𝑘𝑇])as ‖𝑢‖_{𝐶[−𝑘𝑇,𝑘𝑇]}=max{|𝑢(𝑡)|∶𝑡∈[−𝑘𝑇,𝑘𝑇]}.

Consider𝐽_(𝑘)∶𝐸_(𝑘)×[0,+∞)→𝑅^𝑛defined by 𝐽_(𝑘)(𝑢,𝜆)=𝜙₁(𝑢)+𝜆𝜙₂(𝑢), (13) where 𝜙₁(𝑢)=‖𝑢‖^𝑝𝑝,𝜙₂(𝑢)=−^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)((𝑡𝐹𝑡,𝑢))𝑑𝑡,_(𝑘)(𝑡,𝑥)=^𝑥₀𝑓_(𝑘)𝑡,𝑦𝑑𝑦∀(𝑡,𝑥)∈[−𝑘𝑇,𝑘𝑇]×𝑅^𝑛. (14) Using the continuity of 𝑓_(𝑘), one has that 𝐽_(𝑘)(𝑢,𝜆)is (strongly) continuous in𝐸_(𝑘)×[0,+∞), 𝐽_(𝑘)(⋅,𝜆)∈𝐶¹(𝐸_(𝑘),𝑅^𝑛)and for any𝑢,𝑣∈𝐸_(𝑘), 𝐽^′_(𝑘)𝑢(𝑢,𝜆),𝑣=^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)𝑣^′(𝑡)+𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑠(𝑡)Φ_𝑝(𝑢(𝑡))𝑣(𝑡)𝑑𝑡−𝜆^𝑘𝑇_−𝑘𝑇𝑓_(𝑘)(𝑡,𝑢(𝑡))𝑣(𝑡)𝑑𝑡. (15)

In order to prove our main result, we list some basic facts in this section.

Definition 1. A function 𝑢∈𝑢∈𝐸_(𝑘)∶𝜌Φ_𝑝𝑢^′(⋅)∈𝑊^1,∞_2𝑘𝑇𝑅,𝑅^𝑛 (16) is said to be a 2𝑘𝑇-periodic solution of 𝑃_𝑘 if𝑢satisfies the equation in 𝑃_𝑘.

Lemma 2. If𝑢∈𝐸_(𝑘)is a critical point of𝐽_(𝑘)(⋅,𝜆); then𝑢is a 2𝑘𝑇-periodic solution of  𝑃_𝑘.

Proof. Assume that𝑢∈𝐸_(𝑘)is a critical point of𝐽_(𝑘)(⋅,𝜆); then for all𝑣∈𝐸_(𝑘),  one has 0=^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)𝑣^′(𝑡)+𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑠(𝑡)Φ_𝑝(𝑢(𝑡))𝑣(𝑡)𝑑𝑡−𝜆^𝑘𝑇_−𝑘𝑇𝑓_(𝑘)(𝑡,𝑢(𝑡))𝑣(𝑡)𝑑𝑡. (17) It follows that ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡=−^𝑘𝑇_−𝑘𝑇𝑠(𝑡)Φ_𝑝(𝑢(𝑡))𝑣(𝑡)𝑑𝑡+𝜆^𝑘𝑇_−𝑘𝑇𝑓_(𝑘)(𝑡,𝑢(𝑡))𝑣(𝑡)𝑑𝑡. (18) By the definition of weak derivative, (18) implies that 𝜌(𝑡)Φ_𝑝𝑢^′(𝑡)^′=𝑠(𝑡)Φ_𝑝(𝑢(𝑡))−𝜆𝑓(𝑡,𝑢(𝑡)). (19) Thus𝜌Φ_𝑝(𝑢^′)(⋅)∈𝑊^1,∞_2𝑘𝑇(𝑅,𝑅^𝑛)and𝑢satisfies the 𝑃_𝑘. Therefore,𝑢is a solution of 𝑃_𝑘.

Lemma 2 motivates us to apply three critical points theorem to discuss the multiplicity of the 2𝑘𝑇-periodic solution of 𝑃_𝑘. Here, at the end of this section, let us recall some important facts.

Definition 3. Let𝑋be a Banach space and𝑓∶𝑋→(−∞,+∞].𝑓is said to be sequentially weakly lower semi-continuous ifliminf_𝑘→+∞𝑓(𝑥_𝑘)≥𝑓(𝑥) as 𝑥_𝑘⇀𝑥 in𝑋.

Definition 4. Suppose𝐸is a real Banach space. For𝜙∈𝐶¹(𝐸,𝑅^𝑛), we say that 𝜙satisfies PS condition if any sequence{𝑢_𝑘}⊂𝐸for which𝜙(𝑢_𝑘)is bounded and𝜙^′(𝑢_𝑘)→0as𝑘→∞possesses a convergent subsequence.

Lemma 5 (see [16]). For𝑢∈𝐸_(𝑘), one then has‖𝑢‖_{𝐶[−𝑘𝑇,𝑘𝑇]}≤𝑀‖𝑢‖_𝐸(𝑘), where 𝑀=2^1/𝑞1max(2𝑘𝑇)^1/𝑝essinf_{[−𝑘𝑇,𝑘𝑇]}𝑠^1/𝑝,(2𝑘𝑇)^1/𝑞essinf_{[−𝑘𝑇,𝑘𝑇]}𝜌^1/𝑝,1𝑝+1𝑞=1. (20)

Lemma 6 (see [27]). Let𝑋be a nonempty set, andΦ,Ψare two real functions on𝑋. Assume that there are𝑟>0, 𝑥₀,𝑥₁∈𝑋such that Φ𝑥₀=Ψ𝑥₀=0,Φ𝑥₁>𝑟,sup_{𝑥∈Φ−1(]−∞,𝑟}Ψ(𝑥)Ψ𝑥<𝑟¹Φ𝑥₁. (21) Then, for each𝜌satisfying sup_{𝑥∈Φ−1(]−∞,𝑟}Ψ(𝑥)Ψ𝑥<𝜌<𝑟¹Φ𝑥₁, (22) one has sup_𝜆≥0inf_𝑥∈𝑋Φ(𝑥)+𝜆𝜌−Ψ(𝑥)<inf_𝑥∈𝑋sup_𝜆≥0Φ(𝑥)+𝜆𝜌−Ψ(𝑥). (23)

Lemma 7 (see [17]). Let𝑋be a separable and reflexive real Banach space,𝐼⊆𝑅an interval, and𝑓∶𝑋×𝐼→𝑅a function satisfying the following conditions: (i)for each𝑥∈𝑋, the function𝑓(𝑡,⋅)is continuous and concave; (ii)for each𝜆∈𝐼, the function𝑓(𝑡,⋅)is sequentially weakly lower semicontinuous and Dâteaux differentiable, andlim_{‖𝑥‖→∞}𝑓(𝑥,𝜆)=+∞; (iii)there exists a continuous concave functionℎ∶𝐼→𝑅such that sup_𝜆∈𝐼inf_𝑥∈𝑋𝑓(𝑥,𝜆)+ℎ(𝜆)<inf_𝑥∈𝑋sup_𝜆∈𝐼𝑓(𝑥,𝜆)+ℎ(𝜆). (24)Then, there exist an open interval𝐽⊆𝐼and a positive real number𝜌, such that, for each𝜆∈𝐽, the equation 𝑓^′_𝑥(𝑥,𝜆)=0 (25) has at least two solutions in𝑋whose norms are less than𝜌. If, in addition, the function𝑓is (strongly) continuous in𝑋×𝐼, and, for each𝜆∈𝐼, the function𝑓(𝑡,⋅)is𝐶¹and satisfies the PS condition, then the above conclusion holds with “three” instead of “two.”

Lemma 8. Let𝑢∈𝑊^1,𝑝(𝑅,𝑅^𝑛). Then for every𝑡∈𝑅, the following inequality holds: |𝑢(𝑡)|≤^𝑡+1/2_𝑡−1/2|𝑢(𝑠)|^𝑝𝑑𝑠^1/𝑝+12^𝑡+1/2_𝑡−1/2𝑢^′(𝑠)^𝑝𝑑𝑠^1/𝑝. (26)

Proof. Fix𝑡∈𝑅. For every𝜏∈𝑅, |𝑢(𝑡)|≤|𝑢(𝜏)|+^𝑡_𝜏𝑢^′(𝑠)𝑑𝑠. (27) Integrating (27) over[𝑡−1/2,𝑡+1/2]and using the Hölder inequality, we get |𝑢(𝑡)|≤^𝑡+1/2_𝑡−1/2|𝑢(𝜏)|+^𝑡_𝜏𝑢^′(𝑠)≤𝑑𝑠𝑑𝜏_{𝑡+1/2𝑡−1/2}|𝑢(𝜏)|𝑑𝜏+^𝑡_𝑡−1/2^𝑡_𝑡−1/2𝑢^′(𝑠+)𝑑𝑠𝑑𝜏_{𝑡𝑡+1/2𝑡𝑡+1/2}𝑢^′(𝑠≤)𝑑𝑠𝑑𝜏_{𝑡+1/2𝑡−1/2}|𝑢(𝑠)|^𝑝𝑑𝑠^1/𝑝+12^𝑡+1/2_𝑡−1/2𝑢^′(𝑠)^𝑝𝑑𝑠^1/𝑝. (28)

3. Main Result

In this section, our main result of this paper is presented. First, we introduce the following three conditions:(V1) there exist constants𝑐₁,𝛿₁,𝛿₂,𝜂₁>0and𝜂₂>0,  with𝛿²₁+𝛿²₂≠0, 𝜂₁+𝜂₂<𝜂₁𝜂₂  and 𝑐0<¹𝑀<𝐾₂^1/𝑝 (29) such that 2𝑘𝑇  max_{(𝑡,𝑥)∈[−𝑘𝑇,𝑘𝑇]×[−𝑐1,𝑐1]}𝐹_(𝑘)(𝑡,𝑥)<𝐸Ω, where 𝑐𝐸=¹/𝑀^𝑝𝐾₂+𝐾^𝑝₃∫^𝑘𝑇_−𝑘𝑇𝑠(𝑡),𝑑𝑡Ω=^{−𝑘𝑇+2𝑘𝑇/𝜂1}_−𝑘𝑇𝐹_(𝑘)𝑡,𝑔₁(𝑡)+𝑑𝑡_{𝑘𝑇−2𝑘𝑇/𝜂2−𝑘𝑇+2𝑘𝑇/𝜂1}𝐹_(𝑘)𝑡,𝑔₂(𝑡)+𝑑𝑡_{𝑘𝑇𝑘𝑇−2𝑘𝑇/𝜂2}𝐹_(𝑘)𝑡,𝑔₃(𝑡𝐾)𝑑𝑡,₁=𝛿₁𝜂₂+𝛿₂𝜂₁𝜂₁+𝜂₂−𝜂₁𝜂₂,𝐾₂=𝛿₁^{𝑝−𝑘𝑇+2𝑘𝑇/𝜂1}_−𝑘𝑇𝜌(𝑡)+𝐾𝑑𝑡_{1𝑝𝑘𝑇−2𝑘𝑇/𝜂2−𝑘𝑇+2𝑘𝑇/𝜂1}𝜌(𝑡)+𝛿𝑑𝑡_{2𝑝𝑘𝑇𝑘𝑇−2𝑘𝑇/𝜂2}𝜌(𝑡)𝐾𝑑𝑡,𝑘∈𝑁,₃=max2𝑘𝑇𝜂₁𝛿₁,2𝑘𝑇𝜂₂𝛿₂,𝑔₁(𝑡)=𝛿₁(),𝑔𝑡+𝑘𝑇₃(𝑡)=𝛿₂(),𝑔𝑡−𝑘𝑇₂(𝑡)=𝐾₁𝑡+𝑘𝑇−2𝑘𝑇𝜂₁,𝑘∈𝑁; (30) (V2) there exist constant𝜇∈[0,𝑝)and functions𝜏₁(𝑡),𝜏₂(𝑡)∈𝐿([−𝑘𝑇,𝑘𝑇])with essinf_{[−𝑘𝑇,𝑘𝑇]}𝜏₁>0such that 𝐹_(𝑘)(𝑡,𝑥)≤𝜏₁(𝑡)|𝑥|^𝜇+𝜏₂(𝑡)∀𝑡∈[−𝑘𝑇,𝑘𝑇]and𝑥∈𝑅^𝑛; (31) (V3)𝜌,𝑠∈𝐿^∞and𝑓∶𝑅×𝑅^𝑛→𝑅^𝑛 are continuous functions.

Remark 9. If there exist constant𝜇∈[0,𝑝)and functions𝜏₃(𝑡)∈𝐶([−𝑘𝑇,𝑘𝑇])withmin_{[−𝑘𝑇,𝑘𝑇]}𝜏₃>0such that limsup_|𝑥|→∞𝐹_(𝑘)(𝑡,𝑥)|𝑥|^𝜇<𝜏₃(𝑡)uniformly∀𝑡∈[−𝑘𝑇,𝑘𝑇], (32) then (V2) holds.
In fact, (32) implies that there exists𝑐₂>0such that 𝐹_(𝑘)(𝑡,𝑥)≤𝜏₃(𝑡)|𝑥|^𝜇∀𝑡∈[−𝑘𝑇,𝑘𝑇],|𝑥|≥𝑐₂, (33) which combining the continuity of 𝐹_(𝑘)(𝑡,𝑥)−𝜏₃(𝑡)|𝑥|^𝜇on[−𝑘𝑇,𝑘𝑇]×[−𝑐₂,𝑐₂]yields that there exists constant𝑐₃>0such that 𝐹_(𝑘)(𝑡,𝑥)≤𝜏₃(𝑡)|𝑥|^𝜇+𝑐₃∀𝑡∈[−𝑘𝑇,𝑘𝑇],𝑥∈𝑅^𝑛. (34)

Lemma 10. Assume that (V1) holds; then, for each𝑘∈𝑁, there exists a continuous concave functionℎ_(𝑘)∶[0,+∞)→𝑅^𝑛such that sup_𝜆≥0inf_{𝑢∈𝐸(𝑘)}𝐽_(𝑘)(𝑢,𝜆)+ℎ(𝜆)<inf_{𝑢∈𝐸(𝑘)}sup_𝜆≥0𝐽_(𝑘)(𝑢,𝜆)+ℎ(𝜆). (35)

Proof. We define 1𝑟=𝑝𝑐₁𝑀^𝑝,𝑢₁(𝑡)=𝑔₁(𝑡),𝑡∈−𝑘𝑇,−𝑘𝑇+2𝑘𝑇𝜂₁,𝑔₂(𝑡),𝑡∈−𝑘𝑇+2𝑘𝑇𝜂₁,𝑘𝑇−2𝑘𝑇𝜂₂,𝑔₃(𝑡),𝑡∈𝑘𝑇−2𝑘𝑇𝜂₂,𝑘𝑇. (36) It is clear that𝑢₁∈𝐸_(𝑘). It follows from ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢^′₁(𝑡)^𝑝𝑑𝑡=𝐾₂,0≤^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑢₁(𝑡)^𝑝𝑑𝑡≤𝐾^𝑝₃^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑑𝑡 (37) that 𝐾₂≤𝑢₁^𝑝≤𝐾₂+𝐾^𝑝₃^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑑𝑡. (38)
Let𝑔(𝑥)=(1/𝑝)𝑥^𝑝, 𝑥≥0. It is clear that𝑔(𝑥)has the following properties: (1)𝑔(𝑥) strictly increases for𝑥≥0 and (2)𝑔(𝑥)=𝑤has unique solution𝑄(𝑤)for each𝑤>0.
In view of (29), (38), and (1), one has 1𝑝𝑢₁^𝑝≥1𝑝𝐾₂>1𝑝𝑐₁𝑀^𝑝=𝑟>0, (39) which yields that 𝜙₁𝑢₁=1𝑝𝑢₁^𝑝>𝑟>0. (40) It follows from Lemma 5, (1), and (2) that 𝜙₁⁻¹(−∞,𝑟]⊆𝑢∈𝐸_(𝑘)∶1𝑝𝑢₁^𝑝⊆≤𝑟𝑢∈𝐸_(𝑘)∶‖𝑢‖(𝑟⊆≤𝑄)𝑢∈𝐸_(𝑘)∶max_{[]𝑡∈−𝑘𝑇,𝑘𝑇}|𝑢(𝑡)|≤𝑀𝑄(𝑟),𝑘∈𝑁. (41) Let 𝐺=𝑀𝑄(𝑟); then 𝐺/𝑀 is a solution of𝑔(𝑥)=𝑟.From the definition of𝑔(𝑥)and𝑟, we have𝑔(𝑐₁/𝑀)=𝑟. Thus, (2) implies𝐺=𝑐₁, which combining (41) yields that 𝜙₁⁻¹(−∞,𝑟]⊆𝑢∈𝐸_(𝑘)∶max_{[]𝑡∈−𝑘𝑇,𝑘𝑇}|𝑢(𝑡)|≤𝑐₁,𝑘∈𝑁. (42)
Therefore, sup_{𝑢∈𝜙1−1(]−∞,𝑟}−𝜙₂(𝑢)=sup_{𝑢∈𝜙1−1(]−∞,𝑟}^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)(𝑡,𝑢(𝑡))𝑑𝑡≤sup_{|𝑢(𝑡)|≤𝑐1}^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)(𝑡,𝑢(𝑡))𝑑𝑡≤2𝑘𝑇max_{()∈[]×𝑡,𝑥−𝑘𝑇,𝑘𝑇−𝑐1,𝑐1}𝐹_(𝑘)(𝑡,𝑥),𝑘∈𝑁. (43)
Since𝐹_(𝑘)(𝑡,0)=0, we obtain 2𝑘𝑇max_{()∈[]×𝑡,𝑥−𝑘𝑇,𝑘𝑇−𝑐1,𝑐1}𝐹_(𝑘)(𝑡,𝑥)≥0,𝑘∈𝑁. (44) It follows from𝛿²₁+𝛿²₂≠0 that𝐾₂+𝐾^𝑝₃∫^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑑𝑡>0, which combining𝑐₁,𝑀>0yields that𝐸>0. Therefore, in view of (V1) and (44), we getΩ>0. Thus, it follows from (38) and (40) that 𝑟∫^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)𝑡,𝑢₁(𝑡)𝑑𝑡𝜙₁𝑢₁≥𝑟Ω1/𝑝𝑢₁^𝑝≥Ω𝑐₁/𝑀^𝑝𝐾₂+𝐾^𝑝₃∫^𝑘𝑇_−𝑘𝑇𝑠(𝑡),𝑑𝑡𝑘∈𝑁. (45)
From (43), (45), and (V1), we have sup_{𝑢∈𝜙1−1(]−∞,𝑟}−𝜙₂(𝑢)<𝑟−𝜙₂𝑢₁𝜙₁𝑢₁. (46) It is obvious that𝜙₁(0)=−𝜙₂(0)=0. Owing to Lemma 6, choosingℎ(𝜆)=𝜌𝜆, we obtain sup_𝜆≥0inf_{𝑢∈𝐸(𝑘)}𝜙₁(𝑢)+𝜆𝜙₂(𝑢)+ℎ(𝜆)<inf_{𝑢∈𝐸(𝑘)}sup_𝜆≥0𝜙₁(𝑢)+𝜆𝜙₂(𝑢)+ℎ(𝜆), (47) which combining 𝐽_(𝑘)(𝑢,𝜆)=𝜙₁(𝑢)+𝜆𝜙₂(𝑢) implies the conclusion.

Lemma 11. If (V2) holds, then for each𝑘∈𝑁,lim_{‖𝑢‖→∞}𝐽_(𝑘)(𝑢,𝜆)=+∞and𝐽_(𝑘)(⋅,𝜆)satisfies the𝑃𝑆condition.

Proof. Let{𝑢_𝑛^(𝑘)}be a sequence in𝐸_(𝑘)such thatlim_𝑛→+∞𝐽^′_(𝑘)𝑢(𝑢_𝑛^(𝑘),𝜆)=0and𝐽^′_(𝑘)(𝑢_𝑛^(𝑘),𝜆)is bounded, for each𝑘∈𝑁.
Lemma 5 implies that |𝑢(𝑡)|≤‖𝑢‖^∞_{𝐶[−𝑘𝑇,𝑘𝑇]}≤𝑀‖𝑢‖_𝐸(𝑘)∀𝑡∈[−𝑘𝑇,𝑘𝑇]. (48) It follows from (V2) and (48) that ^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)(𝑡,𝑢(𝑡))𝑑𝑡≤^𝑘𝑇_−𝑘𝑇𝜏₁(𝑡)|𝑢(𝑡)|^𝜇𝑑𝑡+^𝑘𝑇_−𝑘𝑇𝜏₂(𝑡)𝑑𝑡≤𝑀^𝜇‖𝑢(𝑡)‖^𝜇𝑘𝑇_−𝑘𝑇𝜏₁(𝑡)𝑑𝑡+^𝑘𝑇_−𝑘𝑇𝜏₂(𝑡)𝑑𝑡, (49) which yields that 𝐽_(𝑘)()≥1𝑢,𝜆𝑝‖𝑢‖^𝑝−𝜆𝑀^𝜇‖𝑢(𝑡)‖^𝜇𝑘𝑇_−𝑘𝑇𝜏₁(𝑡)𝑑𝑡−𝜆^𝑘𝑇_−𝑘𝑇𝜏₂(𝑡)𝑑𝑡, (50) for each𝑘∈𝑁. Noting that𝜇∈[0,𝑝), the above inequality implies thatlim_{‖𝑢‖→∞}𝐽_(𝑘)(𝑢,𝜆)=+∞and{𝑢_𝑛^(𝑘)}is bounded in𝐸_(𝑘). Next, we will prove that{𝑢_𝑛^(𝑘)}converges strongly to some𝑢^(𝑘)in𝐸_(𝑘). The proof is similar to [22]. Since{𝑢_𝑛^(𝑘)}is bounded in𝐸_(𝑘), there exists a subsequence of{𝑢_𝑛^(𝑘)}(for simplicity denoted again by{𝑢_𝑛^(𝑘)}) such that{𝑢_𝑛^(𝑘)}converges weakly to some𝑢^(𝑘)in𝐸_(𝑘). Then{𝑢_𝑛^(𝑘)}converges uniformly to𝑢^(𝑘)on[−𝑘𝑇,𝑘𝑇](see [28]). Therefore, ^𝑘𝑇_−𝑘𝑇𝑓_(𝑘)𝑡,𝑢_𝑛^(𝑘)(𝑡)−𝑓_(𝑘)𝑡,𝑢^(𝑘)(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)𝑑𝑡⟶0 (51) as𝑛→+∞,  for each𝑘∈𝑁. In view thatlim_𝑛→+∞𝐽^′_(𝑘)𝑢(𝑢_𝑛^(𝑘),𝜆)=0 and{𝑢_𝑛^(𝑘)}converges weakly to some𝑢^(𝑘), we get 𝐽^′_(𝑘)𝑢𝑢_𝑛^(𝑘),𝜆−𝐽^′_(𝑘)𝑢𝑢^(𝑘),𝜆,𝑢_𝑛^(𝑘)−𝑢^(𝑘)⟶0 (52) as 𝑛→+∞, for each𝑘∈𝑁. Then, from (15), one has 𝐽^′_(𝑘)𝑢𝑢_𝑛^(𝑘),𝜆−𝐽^′_(𝑘)𝑢𝑢^(𝑘),𝜆,𝑢_𝑛^(𝑘)−𝑢^(𝑘)=^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢_𝑛^′(𝑘)(𝑡)−Φ_𝑝𝑢^′(𝑘)(𝑡)×𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)+𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑠(𝑡)Φ_𝑝𝑢_𝑛^(𝑘)(𝑡)−Φ_𝑝𝑢^(𝑘)(𝑡)×𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)𝑑𝑡−𝜆^𝑘𝑇_−𝑘𝑇𝑓_(𝑘)𝑡,𝑢_𝑛^(𝑘)(𝑡)−𝑓_(𝑘)𝑡,𝑢^(𝑘)(𝑡)×𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)𝑑𝑡 (53) for each𝑘∈𝑁. By [29], for each𝑘∈𝑁, there exist𝑐_𝑝,𝑑_𝑝>0such that ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢_𝑛^′(𝑘)(𝑡)−Φ_𝑝𝑢^′(𝑘)(𝑡)×𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)+𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑠(𝑡)Φ_𝑝𝑢_𝑛^(𝑘)(𝑡)−Φ_𝑝𝑢^(𝑘)(𝑡×𝑢_𝑛(𝑘))(𝑡)−𝑢^(𝑘)(𝑡)≥𝑐𝑑𝑡_𝑝∫^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)^𝑝+𝑠(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)^𝑝𝑑𝑡,if𝑑𝑝≥2,_𝑝∫^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)²𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^2−𝑝+𝑠(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)²𝑢_𝑛^(𝑘)(𝑡)+𝑢^(𝑘)(𝑡)^2−𝑝𝑑𝑡,if1<𝑝<2. (54)
If𝑝≥2, it follows from (51)–(54) that ‖𝑢_𝑛^(𝑘)−𝑢^(𝑘)‖→0 as 𝑛→+∞.
If1<𝑝<2, by Holder’s inequality, we obtain ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)^𝑝≤𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)²𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^2−𝑝𝑑𝑡^𝑝/2×^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^𝑝𝑑𝑡^(2−𝑝)/2≤2^{(𝑝−1)(2−𝑝)/2}×^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)²𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^2−𝑝𝑑𝑡^𝑝/2×^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^𝑝𝑑𝑡^(2−𝑝)/2≤2^{(𝑝−1)(2−𝑝)/2}×^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)²𝑢_𝑛^′(𝑘)(𝑡)+𝑢^′(𝑘)(𝑡)^2−𝑝𝑑𝑡^𝑝/2×𝑢_𝑛^(𝑘)+𝑢^{(𝑘)(2−𝑝)𝑝/2} (55) for each𝑘∈𝑁. Similarly, ^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)^𝑝𝑑𝑡≤2^{(𝑝−1)(2−𝑝)/2𝑘𝑇}_−𝑘𝑇𝑠(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)²𝑢_𝑛^(𝑘)(𝑡)+𝑢^(𝑘)(𝑡)^2−𝑝𝑑𝑡^𝑝/2×𝑢_𝑛^(𝑘)+𝑢^{(𝑘)(2−𝑝)𝑝/2}. (56)
It follows from1<𝑝<2 and (54)–(56) that ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)Φ_𝑝𝑢^′_𝑛^(𝑘)(𝑡)−Φ_𝑝𝑢^′(𝑘)(𝑡)×𝑢^′_𝑛^(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)+𝑑𝑡_{𝑘𝑇−𝑘𝑇}𝑠(𝑡)Φ_𝑝𝑢_𝑛^(𝑘)(𝑡)−Φ_𝑝𝑢^(𝑘)(𝑡×𝑢_𝑛(𝑘))(𝑡)−𝑢^(𝑘)(𝑡)≥2𝑑𝑡_{(𝑝−1)(2−𝑝)/2}𝑑_𝑝𝑢_𝑛^(𝑘)+𝑢^{(𝑘)2−𝑝}×^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢_𝑛^′(𝑘)(𝑡)−𝑢^′(𝑘)(𝑡)^𝑝𝑑𝑡^2/𝑝+^𝑘𝑇_−𝑘𝑇𝑠(𝑡)𝑢_𝑛^(𝑘)(𝑡)−𝑢^(𝑘)(𝑡)^𝑝𝑑𝑡^2/𝑝≥𝑑_𝑝2^𝑝−2𝑢_𝑛^(𝑘)−𝑢^(𝑘)2𝑢_𝑛^(𝑘)+𝑢^{(𝑘)2−𝑝}. (57) In view of (51)–(53) and (57), we have‖𝑢_𝑛^(𝑘)−𝑢^(𝑘)‖→0as𝑛→+∞,  for each𝑘∈𝑁.
Therefore,{𝑢_𝑛^(𝑘)}converges strongly to𝑢^(𝑘)in𝐸_(𝑘), for each𝑘∈𝑁. Thus, for each𝑘∈𝑁, 𝐽_(𝑘)(⋅,𝜆)satisfies the𝑃𝑆condition.

Lemma 12. Assume that (V1) and (V2) hold; then there exist an open interval Λ⊆[0,+∞)and a positive real number𝜎, such that, for each𝜆∈Λ  and𝑘∈𝑁,  𝑃_𝑘 has at least three 2𝑘𝑇-periodic solutions in𝐸_(𝑘)whose norms are less than𝜎.

Proof. Let{𝑢_𝑛^(𝑘)}be a weakly convergent sequence to 𝑢^(𝑘)in𝐸_(𝑘); then{𝑢_𝑛^(𝑘)}converges uniformly sequence to𝑢^(𝑘)on[−𝑘𝑇,𝑘𝑇]. The continuity and convexity of(1/𝑝)‖𝑢^(𝑘)‖^𝑝 imply that(1/𝑝)‖𝑢^(𝑘)‖^𝑝is sequentially weakly lower continuous [28, Lemma 1.2], for each𝑘∈𝑁, which combining the continuity of𝑓_(𝑘)yields that liminf_𝑛→+∞1𝑝𝑢_𝑛^(𝑘)𝑝−𝜆^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)𝑡,𝑢_𝑛^(𝑘)≥1𝑑𝑡𝑝𝑢^(𝑘)𝑝−𝜆^𝑘𝑇_−𝑘𝑇𝐹_(𝑘)𝑡,𝑢^(𝑘)𝑑𝑡. (58) Hence,𝐽_(𝑘)(⋅,𝜆)is sequentially weakly lower semi-continuous, for each𝑘∈𝑁.
It is obvious that𝐽_(𝑘)(𝑢,⋅)is continuous and concave for each𝑢∈𝐸_(𝑘). In view of Lemmas 10 and 11, it follows from Lemma 7 that there exist an open intervalΛ⊆[0,+∞)and a positive real number𝜎, such that, for each𝜆∈Λ and𝑘∈𝑁, 𝐽_(𝑘)(⋅,𝜆)has at least three critical points in𝐸_(𝑘)whose norms are less than𝜎. Therefore, we can reach our conclusion by using Lemma 2.

Lemma 13. Assume that (V3) holds. Let𝑢_(𝑘)∈𝐸_(𝑘)be one of the three 2𝑘𝑇-periodic solutions of system  𝑃_𝑘 obtained by Lemma 12 for each𝑘∈𝑁. Then there exists a subsequence{𝑢_(𝑘𝑗)}of{𝑢_(𝑘)}_𝑘∈𝑁convergent to a certain𝑢₀∈𝐶¹(𝑅,𝑅^𝑛)in𝐶¹_𝑙𝑜𝑐(𝑅,𝑅^𝑛).

Proof. From Lemma 12, we have 𝑢_{(𝑘)𝐸(𝑘)}<𝜎, (59) which combining Lemma 5 yields that there exists a positive constant𝑀₁independent of𝑘such that 𝑢_{(𝑘)𝐿∞2𝑘𝑇}≤𝑀₁. (60) Thus, we obtain that{𝑢_(𝑘)}_𝑘∈𝑁 is a uniformly bounded sequence. Next, we will show that{𝑢^′_(𝑘)}_𝑘∈𝑁and{𝜌Φ_𝑝(𝑢^′_(𝑘))^′}_𝑘∈𝑁are also uniformly bounded sequences. Since{𝑢(𝑡)_(𝑘)}is a 2𝑘𝑇-periodic solutions of system 𝑃_𝑘 for every𝑡∈[−𝑘𝑇,𝑘𝑇),  we have 𝜌(𝑡)Φ_𝑝𝑢^′_(𝑘)(𝑡)^′=𝑠(𝑡)Φ_𝑝𝑢_(𝑘)(𝑡)−𝜆𝑓_(𝑘)𝑡,𝑢_(𝑘)(𝑡). (61)
By (60), (61), and (V3), we get 𝜌(𝑡)Φ_𝑝𝑢^′_(𝑘)(𝑡)^′≤𝑠(𝑡)Φ_𝑝𝑢_(𝑘)(𝑡)+𝜆𝑓_(𝑘)𝑡,𝑢_(𝑘)(𝑡)≤sup_{|𝑥|0≤𝑡<𝑘𝑇,≤𝑀1}𝑠(𝑡)Φ_𝑝(𝑥)+𝜆sup_{|𝑥|0≤𝑡<𝑘𝑇,≤𝑀1}𝑓(𝑡,𝑥)≡𝑀₂,𝑡∈[−𝑘𝑇,𝑘𝑇), (62) which yields that 𝜌(𝑡)Φ_𝑝𝑢^′_(𝑘)(𝑡)^′_{𝐿∞2𝑘𝑇}≤𝑀₀,𝑘∈𝑁. (63) Then, from (63), (V3), and the definition ofΦ_𝑝(𝑥), we obtain 𝑢_(𝑘)^′′(𝑡)_{𝐿∞2𝑘𝑇}≤𝑀₃,𝑘∈𝑁. (64) For𝑖=−𝑘,−𝑘+1,…,𝑘−1, by the continuity of𝑢^′_(𝑘)(𝑡), we can choose𝑡_𝑖^(𝑘)∈[𝑖𝑇,(𝑖+1)𝑇]such that 𝑢^′_(𝑘)𝑡_𝑖^(𝑘)=1𝑇^(𝑖+1)𝑇_𝑖𝑇𝑢^′_(𝑘)(𝑠)𝑑𝑠=𝑇⁻¹𝑢_(𝑘)((𝑖+1)𝑇)−𝑢_(𝑘)(𝑖𝑇); (65)
it follows that for𝑡∈[𝑖𝑇,(𝑖+1)𝑇], 𝑖=−𝑘,−𝑘+1,…,𝑘−1𝑢^′_(𝑘)(𝑡)=^𝑡_{𝑡𝑖(𝑘)}𝑢_(𝑘)^′′(𝑠)𝑑𝑠+𝑢^′_(𝑘)𝑡_𝑖^(𝑘)≤^(𝑖+1)𝑇_𝑖𝑇𝑢_(𝑘)^′′(𝑠)𝑑𝑠+𝑢^′_(𝑘)𝑡_𝑖^(𝑘)≤𝑀₃𝑇+𝑇⁻¹𝑢_(𝑘)((𝑖+1)𝑇)−𝑢_(𝑘)(𝑖𝑇)≤𝑀₃𝑇+2𝑀₁𝑇⁻¹≡𝑀₄. (66) Consequently, 𝑢^′_(𝑘)(𝑡)_{𝐿∞2𝑘𝑇}≤𝑀₄,𝑘∈𝑁. (67)
Now we prove that the sequences{𝑢_(𝑘)}_𝑘∈𝑁and{𝑢^′_(𝑘)}_𝑘∈𝑁 are uniformly bounded and equicontinuous. In fact, for every𝑘∈𝑁and𝑡₁,𝑡₂∈𝑅, we have by (67) 𝑢_(𝑘)𝑡₁−𝑢_(𝑘)𝑡₂=^𝑡2_𝑡1𝑢^′_(𝑘)(𝑠)≤𝑑𝑠_𝑡2𝑡1𝑢^′_(𝑘)(𝑠)𝑑𝑠≤𝑀₄𝑡₁−𝑡₂. (68) Similarly, from (64), we have 𝑢^′_(𝑘)𝑡₁−𝑢^′_(𝑘)𝑡₂≤𝑀₃𝑡₁−𝑡₂. (69) Then, by application of the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence{𝑢_(𝑘𝑗)}of{𝑢_(𝑘)}_𝑘∈𝑁 and a function𝑢₀such that 𝑢_(𝑘𝑗)⟶𝑢₀,as𝑗⟶∞in𝐶¹_loc𝑅,𝑅^𝑛. (70)
Thus, Lemma 13 is proved.

Lemma 14. Let𝑢₀∈𝐶¹(𝑅,𝑅^𝑛)be determined by Lemma 13. Then𝑢₀is a nontrivial homoclinic solution of system (𝑃).

Proof. The first step is to show that𝑢₀is a solution of system (𝑃). By Lemma 13, one has 𝜌(𝑡)Φ_𝑝𝑢^′_𝑘𝑗(𝑡)^′=𝑠(𝑡)Φ_𝑝𝑢_(𝑘𝑗)(𝑡)−𝜆𝑓_(𝑘𝑗)𝑡,𝑢_(𝑘𝑗)(𝑡), (71) for𝑡∈[−𝑘_𝑗𝑇,𝑘_𝑗𝑇), 𝑗∈𝑁. Take𝑎,𝑏∈𝑅with𝑎<𝑏. There exists𝑗₀∈𝑁such that for all𝑗>𝑗₀one has 𝜌(𝑡)Φ_𝑝𝑢^′_𝑘𝑗(𝑡)^′=𝑠(𝑡)Φ_𝑝𝑢_(𝑘𝑗)(𝑡)−𝜆𝑓𝑡,𝑢_(𝑘𝑗)(𝑡),for𝑡∈[𝑎,𝑏]. (72) Integrating (72) from𝑎to𝑡∈[𝑎,𝑏], we obtain 𝜌(𝑡)Φ_𝑝𝑢^′_𝑘𝑗(𝑡)−𝜌(𝑎)Φ_𝑝𝑢^′_𝑘𝑗(𝑎)=^𝑡_𝑎𝑠(𝑣)Φ_𝑝𝑢_(𝑘𝑗)(𝑣)−𝜆𝑓𝑣,𝑢_(𝑘𝑗)(𝑣)𝑑𝑣, (73) for𝑡∈[𝑎,𝑏]. Since (70) shows that𝑢_(𝑘𝑗)→𝑢₀uniformly on[𝑎,𝑏]and𝑢^′_(𝑘𝑗)→𝑢^′₀uniformly on[𝑎,𝑏]as𝑗→∞. Let𝑗→∞in (73), we get 𝜌(𝑡)Φ_𝑝𝑢^′₀(𝑡)−𝜌(𝑎)Φ_𝑝𝑢^′₀(𝑎)=^𝑡_𝑎𝑠(𝑣)Φ_𝑝𝑢₀(𝑣)−𝜆𝑓𝑣,𝑢₀(𝑣)𝑑𝑣, (74) for𝑡∈[𝑎,𝑏]. Since𝑎and𝑏are arbitrary, (74) yields that𝑢₀is a solution of system (𝑃). It is easy to see that𝑢=0is not a solution of system (𝑃) for𝑓(𝑡,0)≠0and so𝑢₀≠0.
Secondly, we will prove that𝑢₀(𝑡)→0as𝑡→±∞. By (59), we have ^𝑘𝑇_−𝑘𝑇𝜌(𝑡)𝑢^′_(𝑘)(𝑡)^𝑝+𝑠(𝑡)𝑢_(𝑘)(𝑡)^𝑝𝑑𝑡≤𝜎^𝑝,𝑘∈𝑁. (75) For every𝑙∈𝑁, there exists𝑗₁∈𝑁such that for𝑗>𝑗₁^𝑙𝑇_−𝑙𝑇𝜌(𝑡)𝑢^′_𝑘𝑗(𝑡)^𝑝+𝑠(𝑡)𝑢_(𝑘𝑗)(𝑡)^𝑝𝑑𝑡≤𝜎^𝑝. (76) Let𝑗→∞in the above and use (70), and it follows that for each𝑙∈𝑁, ^𝑙𝑇_−𝑙𝑇𝜌(𝑡)𝑢^′₀(𝑡)^𝑝+𝑠(𝑡)𝑢₀(𝑡)^𝑝𝑑𝑡≤𝜎^𝑝. (77) Let𝑙→∞in the above, and we get ^∞_−∞𝜌(𝑡)𝑢^′₀(𝑡)^𝑝+𝑠(𝑡)𝑢₀(𝑡)^𝑝𝑑𝑡≤𝜎^𝑝. (78) Thus _{|𝑡|≥𝑟}𝜌(𝑡)𝑢^′₀(𝑡)^𝑝+𝑠(𝑡)𝑢₀(𝑡)^𝑝𝑑𝑡⟶0,as𝑟⟶∞. (79) Combining the above with (V3) we have _{|𝑡|≥𝑟}𝑢^′₀(𝑡)^𝑝+𝑢₀(𝑡)^𝑝𝑑𝑡⟶0,as𝑟⟶∞. (80) By (26), we obtain 𝑢₀(𝑡)≤𝑝^{1/𝑝𝑡+1/2}_𝑡−1/2𝑢₀(𝑠)^𝑝+𝑢^′₀(𝑠)^𝑝𝑑𝑠^1/𝑝. (81) Combining (80) with (81), we get𝑢₀(𝑡)→0as𝑡→±∞.
Finally, we show that 𝑢^′₀(𝑡)⟶0as𝑡⟶±∞. (82) From (60) and (70), one has 𝑢₀(𝑡)≤𝑀₁,for𝑡∈𝑅. (83) From this and (64), we have 𝑢₀^′′(𝑡)≤𝑀₃,for𝑡∈𝑅. (84) If (82) does not hold, then there exist𝜀₀∈(0,1/2)and a sequence{𝑡_𝑘}such that 𝑡₁<𝑡₂<𝑡₃<⋯<𝑡_𝑘<𝑡_𝑘+1𝑢,𝑘=1,2,…,_′0𝑡_𝑘≥2𝜀₀,𝑘=1,2,…, (85) which yield that for𝑡∈[𝑡_𝑘,𝑡_𝑘+𝜀₀/(1+𝑀₃)]𝑢^′₀(𝑡)=𝑢^′₀𝑡_𝑘+^𝑡_𝑡𝑘𝑢₀^′′(𝑠)≥𝑢𝑑𝑠_′0𝑡_𝑘−^𝑡_𝑡𝑘𝑢₀^′′(𝑠)𝑑𝑠≥𝜀₀. (86) It follows that ^∞_−∞𝑢^′₀(𝑡)^𝑝𝑑𝑡≥^∞_𝑘=1^{𝑡𝑘+𝜀0/(1+𝑀3)}_𝑡𝑘𝑢^′₀(𝑡)^𝑝𝑑𝑡=∞, (87) which contradicts to (78) and so (82) holds. The proof is completed.

Lemmas 13 and 14 imply that the limit of the 2𝑘𝑇-periodic solutions of system 𝑃_𝑘 is a nontrivial homoclinic solution of system (𝑃). Combining this with Lemma 10–Lemma 12, we can get the following.

Theorem 15. Assume that (V1), (V2), and (V3) hold. Then system (𝑃) possesses three nontrivial homoclinic solutions.

4. Example

Example 1. Consider the following 𝑝-Laplacian problem: (𝑡+3)Φ₃𝑢^′(𝑡)^′−(2𝑡+2)Φ₃(𝑢(𝑡((𝑡))+𝜆𝑓𝑡,𝑢))=0, (88) where𝜆∈[0,+∞),𝑘𝑇=2, and 𝑓_(𝑘)(𝑡,𝑥)=𝑡𝑥+1,∀(𝑡,𝑥)∈[−2,2]×(−∞,+∞). (89) It is obvious that (V3) holds and for every𝑡∈[−2,2],𝐹_(𝑘)()=𝑡𝑡,𝑥2𝑥²+𝑥−2,∀(𝑡,𝑥)∈[−2,2]×(−∞,+∞). (90) Then, lim_𝑥→+∞𝐹_(𝑘)(𝑡,𝑥)𝑥²=𝑡2 (91) for each𝑡∈[−2,2].Thus, there exists𝑐₄>0such that 𝐹_(𝑘)(𝑡,𝑥)≤2|𝑥|²∀𝑡∈[−2,2],|𝑥|≥𝑐₄, (92) which combining the continuity of 𝐹_(𝑘)(𝑡,𝑥)−2|𝑥|²on[−2,2]×[−𝑐₄,𝑐₄]yields that there exists constant𝑐₅>0such that 𝐹_(𝑘)(𝑡,𝑥)≤2|𝑥|²+𝑐₅foreach(𝑡,𝑥)∈[−2,2]×[−∞,+∞). (93) Therefore, (V2) is satisfied. Furthermore, in view of Lemma 5,𝑀=4. Let𝜂₁=𝜂₂=4, 𝛿₁=1, 𝛿₂=1, and𝑐₁=(√2−1)/2;  then𝐾₁=0, 𝐾₂=6, 𝐾₃=1, 𝐸=1.112×10⁻³, Ω=1/2, and [2−(−2)]max_{√(𝑡,𝑥)∈[−2,2]×[−(√2−1)/2,(2−1)/2]}𝐹_(𝑘)(𝑡,𝑥)≤0. Thus (V1) is satisfied. Moreover, 𝑓_(𝑘)(𝑡,0)=1≠0. In view of Theorem 15, we have that Example 1 possesses three nontrivial homoclinic solutions.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (nos. 11271371 and 10971229).