Multiplicity of Homoclinic Solutions for a Class of Nonperiodic Fourth-Order Differential Equations with General Perturbation

Abstract

In this paper, we investigate a class of nonperiodic fourth-order differential equations with general perturbation. By using the mountain pass theorem and the Ekeland variational principle, we obtain that such equations possess two homoclinic solutions. Recent results in the literature are generalized and significantly improved.

1. Introduction

In this paper, we consider the following class of fourth-order differential equations: 𝑢⁽⁴⁾+𝑤𝑢^′′(𝑥)𝑢()()+𝑎=𝑓𝑥,𝑢+𝜆𝑔𝑥,𝑢,𝑥∈ℝ, (1) where 𝑤 is a constant, 𝑓∈𝐶(ℝ×ℝ,ℝ), 𝑔∈𝐶(ℝ×ℝ,ℝ), and 𝑎∈𝐶(ℝ,ℝ).

Recently, a lot of attention has been focused on the study of homoclinic and heteroclinic solutions for this problem; see [1–8]. This may be due to its concrete applications, such as physics and mechanics; see [9, 10]. More precisely, Tersian and Chaparova [5] studied periodic case. They obtained nontrivial homoclinic solution by using mountain pass theorem. For nonperiodic case, Li [7] studied the existence of nontrivial homoclinic solution for this class of equations. Sun and Wu [8] studied multiple homoclinic solutions for the following nonperiodic fourth-order equations with a perturbation: 𝑢⁽⁴⁾+𝑤𝑢^′′+𝑎(𝑥)𝑢=𝑓(𝑥,𝑢)+𝜆ℎ(𝑥)|𝑢|^𝑝0−2𝑢,𝑥∈ℝ,1≤𝑝₀<2. (2) Before stating the results of [8] and our results, we introduce some notations. Throughout this paper, we denote by ‖‖_𝑟 the 𝐿^𝑟-norm, 2≤𝑟≤∞. 𝐿^∞(ℝ) is the Banach space of essentially bounded functions equipped with the norm ‖𝑢‖_∞=esssup{|𝑢(𝑥)|∶𝑥∈ℝ}. (3) If we take a subsequence of a sequence {𝑢_𝑛}, we will denote it again by {𝑢_𝑛}.

Theorem 1 (see [8]). For any real number 𝑘≥1, if the following conditions are satisfied,
(𝐴₀) there exists a positive constant 𝑎₁ such that 0<𝑎₁≤𝑎(𝑥)→+∞ as |𝑥|→+∞ and 𝑤≤2√𝑎₁.

(F₁)  𝑓(𝑥,𝑠) is a continuous function on ℝ×ℝ such that 𝑓(𝑥,𝑠)≡0 for all 𝑠<0 and 𝑥∈ℝ and, moreover, there exists 𝑏∈𝐿^∞(ℝ,ℝ⁺) with |𝑏|_∞<𝑐₀𝑆²₂ such that lim_𝑠→0+𝑓(𝑥,𝑠)𝑠^𝑘(𝑥)𝑓()=𝑏uniformlyin𝑥∈ℝ,𝑥,𝑠𝑠^𝑘≥𝑏(𝑥)∀𝑠>0,𝑥∈ℝ, (4) where 𝑐₀ is defined by (12) in Section 2 and 𝑆_𝑟 are the best constants for the embedding of 𝐻²(ℝ) in 𝐿^𝑟(ℝ) for 2≤𝑟<∞.

(F₂) There exists 𝑞∈𝐿^∞(ℝ,ℝ⁺) with |𝑞|_∞>𝑐₀𝑆²₂ such that lim_𝑠→∞𝑓(𝑥,𝑠)𝑠^𝑘=𝑞(𝑥)uniformlyin𝑥∈ℝ. (5)

(F₃) There exist two constants 𝜃,𝑑₀ satisfying 𝜃>2 and 0≤𝑑₀<(𝑐₀𝑆²₂(𝜃−2)/2𝜃) such that 𝐹()−1𝑥,𝑠𝜃𝑓(𝑥,𝑠)𝑠≤𝑑₀𝑠²∀𝑠>0,𝑥∈ℝ, (6) where 𝐹(𝑥,𝑠)=∫^𝑠₀𝑓(𝑥,𝜏)𝑑𝜏, then one has the following results:

(i) if 𝑘=1 and 𝜇<1 with 𝜇=inf_ℝ𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²𝑑𝑥∣𝑢∈𝐻²(ℝ),_ℝ𝑞(𝑥)𝑢(𝑥)²𝑑𝑥=1, (7) then there exists Λ>0 such that, for every 𝜆∈(0,Λ), problem (2) has at least two homoclinic solutions,

(ii) if 𝑘>1, then there exists Λ>0 such that, for every 𝜆∈(0,Λ), problem (2) has at least two homoclinic solutions.

The sublinear perturbation ℎ(𝑥)|𝑢|^𝑝−2𝑢, 1≤𝑝<2 is too strict; for example, 𝑔()=ℎ(𝑥𝑢|𝑥,𝑢)|^1/2,|𝑢|ℎ(𝑥𝑢|<1,)|^1/3,|𝑢|≥1, (8) does not satisfy this perturbation. However, one can see that our results in this paper can also work in this case. In addition, 𝜇 in Theorem 1 was defined in ℝ. Motivated by the above results, in this paper, we consider more general perturbation and the case of 𝜇 defined in local bounded open set Ω⊂ℝ.

Now, we state our main result.

Theorem 2. For any real number 𝑘≥1, if (𝐴), (𝐹₁)--(𝐹₃) and the following conditions are satisfied,
(𝐺₁) there exist a constant 𝑝∈(1,2) and a positive function ℎ∈𝐿^2/(2−𝑝)(ℝ,ℝ⁺) such that 𝑔(𝑥,𝑠)≤𝑝ℎ(𝑥)|𝑠|^𝑝−1, (9)(𝐺₂) there exist a constant 𝜈∈(1,2), a bounded open set Ω⊂ℝ, and a positive function 𝜉∈𝐶(Ω,ℝ⁺) such that |𝐺(𝑥,𝑠)|≥𝜉(𝑥)|𝑠|^𝜈,∀𝑥∈Ω, (10) where 𝐺(𝑥,𝑠)=∫^𝑠₀𝑔(𝑥,𝜏)𝑑𝜏, then one has the following results:
(𝑖) if 𝑘=1 and 𝜇<1 with 𝜇=inf_Ω𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²𝑑𝑥∣𝑢∈𝐻²₀(Ω),_Ω𝑞(𝑥)𝑢(𝑥)²𝑑𝑥=1, (11) then there exists Λ>0 such that, for every 𝜆∈(0,Λ), problem (1) has at least two homoclinic solutions,
(𝑖𝑖) if 𝑘>1, then there exists Λ>0 such that, for every 𝜆∈(0,Λ), problem (1) has at least two homoclinic solutions.

The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we give the proof of our main results.

2. Preliminaries

In order to prove our main results, we first give some properties of space 𝑋 on which the variational setting for problem (1) is defined.

Lemma 3 (see [5]). Assume that the (𝐴) holds. Then, there exists a constant 𝑐₀>0 such that _ℝ𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²𝑑𝑥≥𝑐₀‖𝑢‖²_𝐻2,∀𝑢∈𝐻²(ℝ), (12) where ‖𝑢‖²_𝐻2=(∫_ℝ[𝑢^′′(𝑥)²+𝑢^′(𝑥)²+𝑢(𝑥)²]𝑑𝑥)^1/2 is the norm of Sobolev space 𝐻²(ℝ).

Letting 𝑋=𝑢∈𝐻²(ℝ)∣_ℝ𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²,𝑑𝑥<+∞ (13) then 𝑋 is a Hilbert space with the inner product (𝑢,𝑣)=_ℝ𝑢^′′(𝑥)𝑣^′′(𝑥)−𝑤𝑢^′(𝑥)𝑣^′(𝑥)+𝑎(𝑥)𝑢(𝑥)𝑣(𝑥)𝑑𝑥 (14) and the corresponding norm ‖𝑢‖²=(𝑢,𝑢). Note that 𝑋⊂𝐻²(ℝ)⊂𝐿^𝑟(ℝ) (15) for all 𝑟∈[2,+∞] with the embedding being continuous.

Lemma 4 (see [8]). Assume that the condition (𝐴) holds. Then, 𝑋 is compactly embedded in 𝐿^𝑟(ℝ) for all 𝑟∈[2,+∞].

Now, we begin describing the variational formulation of problem (1). Consider the functional 𝐽∶𝑋→ℝ defined by 𝐽(𝑢)=12‖𝑢‖²−_ℝ𝐹(𝑥,𝑢)𝑑𝑥−𝜆_ℝ𝐺(𝑥,𝑢)𝑑𝑥. (16) Since 𝑓, 𝑔 are continuous, by Lemma 4, 𝐽∈𝐶¹(𝑋,ℝ) and its derivative is given by 𝐽^′(𝑢)𝑣=_ℝ𝑢^′′(𝑥)𝑣^′′(𝑥)−𝑤𝑢^′(𝑥)𝑣^′(𝑥)(𝑥)𝑢(𝑥)𝑣(𝑥−+𝑎)𝑑𝑥_ℝ𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)𝑑𝑥−𝜆_ℝ𝑔(𝑥,𝑢(𝑥))𝑣(𝑥)𝑑𝑥 (17) for all 𝑢,𝑣∈𝑋. In addition, any critical point of 𝐽 on 𝑋 is a classical solution of problem (1).

Next, we give the variant version of the mountain pass theorem which is important for the proof of our main results.

Theorem 5 (see [11]). Let 𝐸 be a real Banach space with its dual space 𝐸^∗, and suppose that 𝐼∈𝐶¹(𝐸,ℝ) satisfies max{𝐼(0),𝐼(𝑒)}≤𝜇<𝜂≤inf_{‖𝑢‖=𝜌}𝐼(𝑢) (18) for some 𝜇<𝜂, 𝜌>0 and 𝑒∈𝐸 with ‖𝑒‖>𝜌. Let 𝑐̂≥𝜂 be characterized by 𝑐̂=inf_𝛾∈Γmax_0≤𝜏≤1𝐼𝛾(𝜏), (19) where Γ={𝛾∈𝐶([0,1],𝐸)∶𝛾(0)=0,𝛾(1)=𝑒} is the set of continuous paths joining 0 and 𝑒; then, there exists a sequence {𝑢_𝑛}⊂𝐸 such that 𝐼𝑢_𝑛⟶𝑐̂≥𝜂,1+𝑢_𝑛𝐼^′𝑢_𝑛𝐸∗⟶0,𝑎𝑠𝑛⟶∞. (20)

3. Proof of the Main Results

To prove our main results, we first give the following lemma.

Lemma 6. For any real number 𝑘≥1, assume that conditions (𝐴), (𝐹₁)-(𝐹₂), and (𝐺) hold. Then, there exists Λ>0 such that for every 𝜆∈(0,Λ) there exist two positive constants 𝜌,𝜂 such that 𝐽(𝑢)|_{‖𝑢‖=𝜌}≥𝜂>0.

Proof. For any 𝜖>0, it follows from conditions (F₁)-(F₂) that there exist 𝐶_𝜖>0 and 𝑟>max{2,𝑘} such that 𝐹()≤|𝑏|𝑥,𝑠^∞+𝜖2𝑠²+𝐶_𝜖𝑟|𝑠|^𝑟,∀𝑠∈ℝ. (21) By (9) and (21), the Sobolev inequality, and the Hölder inequality, one has, for all 𝑢∈𝑋, 𝐽(𝑢)=12‖𝑢‖²−_ℝ𝐹(𝑥,𝑢)𝑑𝑥−𝜆_ℝ𝐺()≥1𝑥,𝑢𝑑𝑥2‖𝑢‖²−_ℝ|𝑏|_∞+𝜖2𝑢(𝑥)²−𝑑𝑥_ℝ𝐶_𝜖𝑟𝑢(𝑥)^𝑟𝑑𝑥−𝜆_ℝℎ(𝑥)|𝑢(𝑥)|^𝑝≥1𝑑𝑥2‖𝑢‖²−|𝑏|_∞+𝜖2𝑐₀𝑆²₂‖𝑢‖²−𝐶_𝜖𝑟𝑐₀^𝑟/2𝑆^𝑟_𝑟‖𝑢‖^𝑟−𝜆𝑐₀^−𝑝/2𝑆₂^−𝑝‖ℎ‖_2/(2−𝑝)‖𝑢‖^𝑝=‖𝑢‖^𝑝12|𝑏|1−^∞+𝜖2𝑐₀𝑆²₂‖𝑢‖^2−𝑝−𝐶_𝜖𝑟𝑐₀^𝑟/2𝑆^𝑟_𝑟‖𝑢‖^𝑟−𝑝−𝜆𝑐₀^−𝑝/2𝑆₂^−𝑝‖ℎ‖_2/(2−𝑝). (22) Take 𝜖=(𝑐₀𝑆²₂/2)−|𝑏|_∞ and define 𝑙(𝑡)=14𝑡^2−𝑝−𝐶_𝜖𝑟⁻¹𝑐₀^−𝑟/2𝑆_𝑟^−𝑟𝑡^𝑟−𝑝,∀𝑡≥0. (23) It is easy to prove that there exists 𝜌>0 such that max_𝑡≥0𝑙(𝑡)=𝑙𝜌=𝑟−24𝑟−𝑝2−𝑝𝑟𝑐₀^𝑟/2𝑆^𝑟_𝑟4𝑟−𝑝𝐶_𝜖^{(2−𝑝)/(𝑟−2)}. (24) Then, it follows from (22) that there exists Λ>0 such that for every 𝜆∈(0,Λ) there exist two positive constants 𝜌, 𝜂 such that 𝐽(𝑢)|_{‖𝑢‖=𝜌}≥𝜂>0.

Consider the minimum problem 𝜇=inf_Ω𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²𝑑𝑥∣𝑢∈𝐻²₀(Ω),_Ω𝑞(𝑥)𝑢(𝑥)²𝑑𝑥=1. (25) Then, we have the following results.

Lemma 7. There exist a constant 𝑐₁>0 and 𝜙₁∈𝐻₀^2,2(Ω) with ∫_Ω𝑞(𝑥)𝜙₁(𝑥)²𝑑𝑥=1 such that 𝜇≥𝑐₁ and 𝜇=_Ω𝜙₁^′′(𝑥)²−𝑤𝜙^′₁(𝑥)²+𝑎(𝑥)𝜙₁(𝑥)²𝑑𝑥; (26) that is, the minimum (25) is achieved.

Proof. For any 𝑢∈𝐻₀^2,2(Ω) with ∫_Ω𝑞(𝑥)𝑢(𝑥)²𝑑𝑥=1, by Lemma 3 and the Sobolev embedded theorem, we have _Ω𝑢^′′(𝑥)²−𝑤𝑢^′(𝑥)²+𝑎(𝑥)𝑢(𝑥)²𝑑𝑥≥𝑐₀‖𝑢‖²_𝐻20≥𝑐₀𝑆²₂‖𝑢‖²₂≥𝑐₀𝑆²₂𝑞_∞>0. (27) Therefore, there exists a constant 𝑐₁>0 such that 𝜇≥𝑐₁. Let {𝑢_𝑛}⊂𝐻²₀(Ω) be a minimizing sequence of (25). Clearly, ∫_Ω𝑞(𝑥)𝑢_𝑛(𝑥)²𝑑𝑥=1 and {𝑢_𝑛} is bounded. Then, there exist a subsequence {𝑢_𝑛} and 𝜙₁∈𝐻²₀(Ω) such that 𝑢_𝑛⇀𝜙₁ weakly in 𝐻²₀(Ω) and 𝑢_𝑛→𝜙₁ strongly in 𝐿²(Ω). So it is easy to verify that ∫_Ω𝑞(𝑥)𝑢_𝑛(𝑥)²𝑑𝑥→∫_Ω𝑞(𝑥)𝜙₁(𝑥)²𝑑𝑥 as 𝑛→∞ and ∫_Ω𝑞(𝑥)𝜙₁(𝑥)²𝑑𝑥=1. Therefore, 𝜇≤_Ω𝜙₁^′′(𝑥)²−𝑤𝜙^′₁(𝑥)²+𝑎(𝑥)𝜙₁(𝑥)²𝑑𝑥≤liminf_𝑛→∞Ω𝑢_𝑛^′′(𝑥)²−𝑤𝑢^′_𝑛(𝑥)²+𝑎(𝑥)𝑢_𝑛(𝑥)²𝑑𝑥≤𝜇. (28) This implies that 𝜇=_Ω𝜙₁^′′(𝑥)²−𝑤𝜙^′₁(𝑥)²+𝑎(𝑥)𝜙₁(𝑥)²𝑑𝑥. (29)

Lemma 8. For any real number 𝑘≥1, assume that conditions (𝐴), (𝐹₁)-(𝐹₂), and (𝐺₁)-(𝐺₂) hold. Let 𝜌,Λ>0 be as in Lemma 6. Then, one has the following results.
(𝑖) If 𝑘=1 and 𝜇<1, then there exists 𝑒∈𝑋 with ‖𝑒‖>𝜌 such that 𝐽(𝑒)<0 for all 𝜆∈(0,Λ).
(𝑖𝑖) If 𝑘>1, then there exists 𝑒∈𝑋 with ‖𝑒‖>𝜌 such that 𝐽(𝑒)<0 for all 𝜆∈(0,Λ).

Proof. (i) In case 𝑘=1. Since 𝜇<1, we can choose a nonnegative function 𝜑∈𝐻²₀(Ω) with ∫_Ω𝑞(𝑥)𝜑(𝑥)²𝑑𝑥=1 such that _Ω𝜑^′′(𝑥)²−𝑤𝜑^′(𝑥)²+𝑎(𝑥)𝜑(𝑥)²𝑑𝑥<1. (30) Therefore, by condition (F₂) and the Fatou lemma, we have lim_𝑡→+∞𝐽𝑡𝜑𝑡²≤12𝜑²−lim_{𝑡→+∞Ω}𝐹𝑥,𝑡𝜑𝑡²𝜑²𝜑²𝑑𝑥−lim_𝑡→+∞𝜆𝑡^2−𝜈_Ω𝜉(𝑥)𝜑^𝜈≤1𝑑𝑥2𝜑²−12_Ω𝑞(𝑥)𝜑²=1𝑑𝑥2𝜑²−1<0. (31) So, 𝐽(𝑡𝜑)→−∞ as 𝑡→+∞; then, there exists 𝑒∈𝑋 with ‖𝑒‖>𝜌 such that 𝐽(𝑒)<0 for all 𝜆∈(0,Λ).
(ii) In case 𝑘>1. 𝑞∈𝐿^∞(ℝ,ℝ⁺) with 𝑞⁺≠0 in Ω, and we can choose a function 𝜔∈𝐻²₀(Ω) such that _Ω𝑞(𝑥)|𝜔|^𝑘+1𝑑𝑥>0. (32) Therefore, by condition (F₂) and the Fatou lemma, we have lim_𝑡→+∞𝐽(𝑡𝜔)𝑡^𝑘+1≤‖𝜔‖²2𝑡^𝑘−1−lim_{𝑡→+∞Ω}𝐹(𝑥,𝑡𝜔)𝑡^𝑘+1𝜔^𝑘+1𝜑^𝑘+1𝑑𝑥−lim_𝑡→+∞𝜆𝑡^{𝑘+1−𝜈}_Ω𝜉(𝑥)|𝜔|^𝜈1𝑑𝑥≤−𝑘+1_Ω𝑞(𝑥)𝜔^𝑘+1𝑑𝑥<0. (33)
So, 𝐽(𝑡𝜔)→−∞ as 𝑡→+∞; then, there exists 𝑒∈𝑋 with ‖𝑒‖>𝜌 such that 𝐽(𝑒)<0 for all 𝜆∈(0,Λ).

Next, we define 𝛽=inf_𝛾∈Γmax_0≤𝜏≤1𝐽𝛾(𝜏), (34) where Γ={𝛾∈𝐶([0,1],𝐸)∶𝛾(0)=0,𝛾(1)=𝑒}. Then, by Theorem 5 and Lemmas 6 and 8, there exists a sequence {𝑢_𝑛}⊂𝑋 such that 𝐽𝑢_𝑛⟶𝛽,1+𝑢_𝑛𝐽^′𝑢_𝑛𝐸∗⟶0,as𝑛⟶∞. (35)

Then, we have the following results.

Lemma 9. For any real number 𝑘≥1, assume that conditions (𝐴), (𝐹₁)--(𝐹₃), and (𝐺₁)-(𝐺₂) hold. Let Λ>0 be as in Lemma 6. Then, {𝑢_𝑛} defined by (35) is bounded in 𝑋 for all 𝜆∈(0,Λ).

Proof. For 𝑛 large enough, by the Hölder inequality and Lemma 3, one has 𝛽+1≥𝐽𝑢_𝑛−1𝜃𝐽^′𝑢_𝑛,𝑢_𝑛=12−1𝜃𝑢_𝑛²−_ℝ𝐹𝑥,𝑢_𝑛−1𝜃𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥−𝜆_ℝ𝐺𝑥,𝑢_𝑛−1𝜃𝑔𝑥,𝑢_𝑛𝑢_𝑛≥𝑑𝑥𝜃−2𝑢2𝜃^𝑛2−𝑑_0ℝ𝑢²_𝑛𝑝𝑑𝑥−𝜆1+𝜃_ℝℎ(𝑥)𝑢^𝑝_𝑛≥𝑑𝑥𝜃−2𝑢2𝜃^𝑛2−𝑑₀𝑐₀𝑆²₂𝑢_𝑛²𝑝−𝜆1+𝜃𝑆₂^−𝑝|ℎ|_2/(2−𝑝)𝑢_𝑛^𝑝≥𝜃−2−𝑑2𝜃𝑐₀𝑆⁰²₂𝑢_𝑛²𝑝−Λ1+𝜃𝑆₂^−𝑝|ℎ|_2/(2−𝑝)𝑢_𝑛^𝑝, (36) which implies that {𝑢_𝑛} is bounded in 𝑋, since 1≤𝑝<2.

For 𝜌 given by Lemma 6, denote 𝐵_𝜌={𝑢∈𝑋∶‖𝑢‖<𝜌}. Then, by Ekeland’s variational principle and Lemma 4, we have the following lemma, which shows that 𝐽 has a local minimum if 𝜆 is small.

Lemma 10. For any real number 𝑘≥1, assume that conditions (𝐴), (𝐹₁)--(𝐹₃), and (𝐺₁)-(𝐺₂) hold. Let Λ>0 be as in Lemma 6. Then, for every 𝜆∈(0,Λ), there exists 𝑢₀∈𝑋 such that 𝐽𝑢₀=inf𝐽(𝑢)∶𝑢∈𝐵_𝜌<0, (37) and 𝑢₀ is a homoclinic solution of problem (1).

Proof. Since 𝜉∈𝐿^∞(ℝ,ℝ⁺) with 𝜉⁺≠0 in Ω, we can choose a function 𝜓∈𝐻²₀(Ω) such that _Ω𝜉(𝑥)𝜓^𝜈𝑑𝑥>0. (38) Hence, we have 𝐽=𝑡𝑡𝜓²2𝜓²−_Ω𝐹𝑥,𝑡𝜓𝑑𝑥−𝜆𝑡^𝜈_Ω𝜉(𝑥)𝜓^𝜈≤𝑡𝑑𝑥₂2𝜓²−𝜆𝑡^𝜈_Ω𝜉(𝑥)𝜓^𝜈𝑑𝑥<0 (39) for 𝑡>0 small enough. This implies 𝜃₀∶=inf{𝐽(𝑢)∶𝑢∈𝐵_𝜌}<0. By the Ekeland variational principle, there exists a minimizing sequence {𝑢_𝑛}⊂𝐵_𝜌 such that 𝐽(𝑢_𝑛)→𝜃₀ and 𝐽^′(𝑢_𝑛)→0 as 𝑛→∞. Hence, Lemma 4 implies that there exists 𝑢₀∈𝑋 such that 𝐽^′(𝑢₀)=0 and 𝐽(𝑢₀)=𝑐₁<0.

Proof of Theorem 2. From Lemmas 4 and 8, there is 𝑢∈𝑋 such that, up to a subsequence, 𝑢_𝑛⇀𝑢 weakly in 𝑋 and 𝑢_𝑛→𝑢 strongly in 𝐿^𝑠(ℝ) for 𝑠∈[2,∞]. By using a standard procedure, we can prove that 𝑢_𝑛→𝑢 strongly in 𝑋. Moreover, 𝐽(𝑢)=𝛽>0 and 𝑢 is another homoclinic solution of problem (1). Therefore, combining with Lemma 10, we prove that problem (1) has at least two homoclinic solutions 𝑢₀,𝑢∈𝑋 satisfying 𝐽(𝑢₀)<0 and 𝐽(𝑢)>0.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The paper was supported by NFSC (11271372).