Weak Solutions for a Second Order Dirichlet Boundary Value Problem on Time Scale

Abstract

We adopt the Leray-Schauder degree theory and critical point theory to consider a second order Dirichlet boundary value problem on time scales and obtain some existence theorems of weak solutions for the previous problem.

1. Introduction

In recent years, differential equations on time scales have been studied extensively in the literature. There have been many approaches to study the existence and multiplicity of solutions for differential equations on time scales. The variational method is, to the best of our knowledge, novel, and it may open a new approach to deal with nonlinear problems on time scales. For more details about recent development in the direction, we refer the reader to [1–12] and the references therein. The two books [13, 14] by Bohner and Peterson summarize some excellent results on time scales.

In [1, 2], the authors utilized variational techniques and critical point theory to derive some sufficient conditions for the existence of positive solutions for the following second order dynamic equation with Dirichlet boundary conditions: −𝑢^ΔΔ(𝑡)=𝑓𝜎(𝑡),𝑢^𝜎(𝑡)𝑢(𝑎)(𝑏).,Δ-a.e.𝑡∈𝐽,=0=𝑢 (1)

In [8], Zhou and Li studied Sobolev's spaces on time scales, and given their properties, as applications, they presented variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian systems on time scales as follows: 𝑢^Δ2(𝑡)=∇𝐹𝜎(𝑡),𝑢^𝜎(𝑡),Δ-a.e.𝑡∈[0,𝑇]^𝜅2_𝕋,𝑢(0)=𝑢(𝑇),𝑢^Δ(0)=𝑢^Δ(𝑇), (2) where 𝐹(𝑡,𝑥) is Δ-measurable in 𝑡 for every 𝑥∈ℝ^𝑁 and continuously differentiable in 𝑥 for Δ-a.e. 𝑡∈[0,𝑇]_𝕋.

Motivated by the works cited previously, in this paper, we study the second order Dirichlet boundary value problem on time scales. Consider −𝑢^ΔΔ(𝑡)=𝜆𝑢^𝜎(𝑡)+ℎ𝜎(𝑡),𝑢^𝜎(𝑡)𝑢(0)(𝑇),Δ-a.e.𝑡∈𝐽,=𝑢=0, (3) where we say that a property holds for Δ-almost every 𝑡∈𝐴⊂𝕋 or Δ-almost everywhere on 𝐴⊂𝕋, Δ-a.e., whenever there exists a set 𝐸⊂𝐴 with null Lebesgue Δ-measure such that this property holds for every 𝑡∈𝐴⧵𝐸, 𝐽∶=[0,𝜌(𝑇))∩𝕋 and 𝕋⊂ℝ is an arbitrary bounded time scale such that min𝕋=0 and max𝕋=𝑇.

We assume that 𝜆 is a parameter, ℎ∈CAR([0,𝑇)_𝕋,ℝ), [0,𝑇)_𝕋∶=[0,𝑇)∩𝕋, where CAR([0,𝑇)_𝕋,ℝ) means that ℎ fulfils the Carathéodory conditions (see [15, Definition 3.2.22] and [2, (i) and (ii) in (H₂)]). By virtue of the Leray-Schauder degree theory, a result involving the existence of weak solutions is established. Next, we utilize two critical point theorems to investigate that problem (3) has at least one weak solution and infinitely many weak solutions with the parameter 𝜆<0, respectively. The results obtained here improve some existing results in the literature.

2. Preliminaries and an Existence Theorem via Leray-Schauder Degree Theory

We first offer some related preliminaries concerning the basic definitions and results on time scales. For convenience, for 𝑓∈𝐿²_Δ([0,𝑇)_𝕋,ℝ), we denote ∫^𝑇₀|𝑓(𝑠)|²Δ𝑠=∫_[0,𝑇)𝕋|𝑓(𝑠)|²Δ𝑠. Let Sobolev's space 𝑊^1,2_Δ,𝑇 be defined by (see [1–3]) 𝑊^1,2_Δ,𝑇∶=𝑥∈𝐴𝐶[0,𝑇]_𝕋,ℝ∶𝑥^Δ∈𝐿²_Δ[0,𝑇)_𝕋,𝑥(0)(𝑇),,ℝ=𝑥=0 (4) where 𝐴𝐶([0,𝑇]_𝕋,ℝ) (see [3, Definition 2.9]) denotes the set of absolutely continuous functions on [0,𝑇]_𝕋. Then 𝑊^1,2_Δ,𝑇 is a Hilbert space with the inner product (𝑢,𝑣)₁=_[0,𝑇)𝕋𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡, (5) and let ‖⋅‖₁ be the norm induced by the inner product (⋅,⋅)₁ (see [1, page 1265] and [2, page 371]).

Lemma 1 (see [3, Proposition 3.7]). The immersion 𝑊^1,2_Δ,𝑇↪𝐶([0,𝑇]_𝕋,ℝ) is compact.
We firstly need to consider the following eigenvalue problem: −𝑢^ΔΔ(𝑡)=𝜆𝑢^𝜎(𝑡),𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢(𝑇)=0. (6) For each 𝑣∈𝑊^1,2_Δ,𝑇, multiply by 𝑣^𝜎(𝑡) on both sides of the previous equation in (6) and integrate over [0,𝑇)_𝕋 to obtain _[0,𝑇)𝕋−𝑢^ΔΔ(𝑡)𝑣^𝜎(𝑡)Δ𝑡=𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡. (7)
By (7) of [8], we obtain _[0,𝑇)𝕋𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡=𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡. (8)

Lemma 2 (see [9, Lemma 3.4] and [13, Theorem 4.95]). The eigenvalues of (6) may be arranged as 0<𝜆₁<𝜆₂<⋯, and 𝜆₁ can be expressed by 𝜆₁∶=inf_{𝑢∈𝑊1,2Δ,𝑇,𝑢≠0}∫_[0,𝑇)𝕋𝑢^Δ(𝑡)²Δ𝑡∫_[0,𝑇)𝕋|𝑢^𝜎(𝑡)|².Δ𝑡 (9) By Lemma 2, if there exists 𝑢₁∈𝑊^1,2_Δ,𝑇 and ‖𝑢₁‖₁=1, then we have 1𝜆₁=_[0,𝑇)𝕋𝑢^𝜎₁(𝑡)²Δ𝑡. (10) Choosing 𝜑^𝜎₁=𝜆₁𝑢^𝜎₁, we can easily obtain _[0,𝑇)𝕋𝜑^𝜎₁(𝑡)²Δ𝑡=𝜆_{1[0,𝑇)𝕋}𝑢^𝜎₁(𝑡)²𝜆Δ𝑡=1,₁=_[0,𝑇)𝕋𝜑^Δ₁(𝑡)²Δ𝑡. (11) Moreover, we also have _[0,𝑇)𝕋𝜑^Δ₁(𝑡)𝑣^Δ(𝑡)Δ𝑡=𝜆_{1[0,𝑇)𝕋}𝜑^𝜎₁(𝑡)𝑣^𝜎(𝑡)Δ𝑡,∀𝑣∈𝑊^1,2_Δ,𝑇. (12)

Lemma 3 (see [16, Corollary 3.3]). Let 𝑢∈𝑊^1,2_Δ,𝑇. Then the Wirtinger-type inequality is _[0,𝑇)𝕋𝑢^𝜎(𝑡)²1Δ𝑡≤𝜆_{1[0,𝑇)𝕋}𝑢^Δ(𝑡)²Δ𝑡, (13) where 𝜆₁ is defined by Lemma 2.
Denote an operator 𝐴∶𝑊^1,2_Δ,𝑇→𝑊^1,2_Δ,𝑇 as follows (𝐴𝑢,𝑣)=_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡,∀𝑢,𝑣∈𝑊^1,2_Δ,𝑇. (14) By Hölders inequality and Lemma 3, we have _[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)≤Δ𝑡_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡≤)Δ𝑡_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡^1/2_[0,𝑇)𝕋𝑣^𝜎(𝑡)²Δ𝑡^1/2≤1𝜆₁‖𝑢‖₁‖𝑣‖₁,∀𝑢,𝑣∈𝑊^1,2_Δ,𝑇. (15) Then 𝐴 is a bounded linear operator. By Lemma 1, 𝐴 is compact. Clearly, 𝐴 is also symmetric, (𝐴𝑢,𝑢)>0 for 𝑢^𝜎≢0. Consequently, the supremum 1/𝜆₁ is achieved by Theorem 6.3.12 in [15]. Hence, (10)–(12) are true.

In what follows, we offer two lemmas involving Leray-Schauder degree.

Lemma 4 (see [15, Proposition 5.2.22]). Let Ω be an open bounded set in a Banach space 𝑋 and 𝐴∈𝒞(Ω,𝑋). Let 𝑥₀∈Ω be a unique solution in Ω of the equation 𝑥=𝐹(𝑥). Assume that the Fréchet derivative 𝐴^′(𝑥₀) exists and 𝐼−𝐴^′(𝑥₀) is continuously invertible. Then deg(𝐼−𝐴,Ω,0)=(−1)^𝛽,𝑤ℎ𝑒𝑟𝑒𝛽=_{𝐹𝜆∈𝜎′𝑥0∩ℝ,𝜆>1}𝑚(𝜆), (16) and 𝑚(𝜆) is the multiplicity of the eigenvalue 𝜆 of the operator 𝐴^′(𝑥₀).

Lemma 5 (see [15, Theorem 5.2.13]). Let Ω be an open bounded set in a Banach space 𝑋. There exists a mapping deg(𝐼−𝐴,Ω,𝑦₀) defined for all 𝐴∈𝒞(Ω,𝑋) and 𝑦₀∈𝑋 such that 𝑥−𝐴(𝑥)≠𝑦₀, ∀𝑥∈𝜕Ω. This mapping has the homotopy invariance property: if 𝐴,𝐵∈𝒞(Ω,𝑋) and 𝐻(𝑡,𝑥)=(1−𝑡)𝐴𝑥+𝑡𝐵𝑥, 𝑡∈[0,1], and 𝑥∈Ω are such that 𝑥−𝐻(𝑡,𝑥)≠𝑦₀, for every 𝑥∈𝜕Ω and 𝑡∈[0,1], then deg(𝐼−𝐴,Ω,𝑦₀)=deg(𝐼−𝐵,Ω,𝑦₀).

Now, we list our hypotheses for (3).(H1)𝜆 is a parameter and 𝜆≠𝜆_𝑛, where 𝜆_𝑛 are determined by Lemma 2.(H2)ℎ∈CAR([0,𝑇)_𝕋,ℝ); there exist 𝑓∈𝐿²_Δ([0,𝑇)_𝕋,ℝ), 𝑐>0 and 𝛼∈[0,1) such that ℎ(𝑡,𝑥)≤𝑓(𝑡)+𝑐|𝑥|^𝛼,∀𝑡∈[0,𝑇)_𝕋,𝑥∈ℝ. (17)(H3)ℎ is 𝛼-Lipschitz continuous with respect to the second variable; that is, there exists a constant 𝑐>0, such that ℎ𝑡,𝑥₁−ℎ𝑡,𝑥₂≤𝑐𝑥₁−𝑥₂^𝛼,∀𝑡∈[0,𝑇)_𝕋,𝑥₁,𝑥₂∈ℝ,𝛼asin(H2). (18)

Remark 6. We can take ℎ(𝑡,𝑥)=𝑓(𝑡)+𝑐|𝑥|^𝛼, where 𝑐,𝑓,𝛼 are as in (H2). Clearly, it also satisfies (H3). In fact, ℎ𝑡,𝑥₁−ℎ𝑡,𝑥₂=𝑐𝑥₁^𝛼−𝑥₂^𝛼≤𝑐𝑥₁−𝑥₂^𝛼,∀𝑥₁,𝑥₂∈ℝ. (19)
It is clear that (3) is equivalent to the following integral equation: _[0,𝑇)𝕋𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡=𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)+Δ𝑡_[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡,∀𝑣∈𝑊^1,2_Δ,𝑇. (20) We define an operator 𝑆∶𝑊^1,2_Δ,𝑇→𝑊^1,2_Δ,𝑇 as follows: (𝑆𝑢,𝑣)=_[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡,∀𝑢,𝑣∈𝑊^1,2_Δ,𝑇. (21)

Lemma 7. 𝑆 is compact on 𝑊^1,2_Δ,𝑇.

Proof. We first prove that there exists a ball 𝐵(0,𝑅)⊂𝑊^1,2_Δ,𝑇 (𝑅>0) such that 𝑆 maps 𝐵(0,𝑅) into itself if 𝑅 is large enough. Indeed, (H2) leads to ‖𝑆𝑢‖₁=sup_{‖𝑣‖1=1}|𝑆𝑢,𝑣|=sup_{‖𝑣‖1=1[0,𝑇)𝕋}ℎ𝜎(𝑡),𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡≤sup_{‖𝑣‖1=1[0,𝑇)𝕋}ℎ𝜎(𝑡),𝑢^𝜎(𝑡)²Δ𝑡^1/2×_[0,𝑇)𝕋𝑣^𝜎(𝑡)²Δ𝑡^1/2≤1𝜆₁^[0,𝑇)_𝕋𝑓^𝜎(𝑡)+𝑐𝑢^𝜎(𝑡)^𝛼2Δ𝑡^1/2≤2𝜆_{1[0,𝑇)𝕋}𝑓^𝜎(𝑡)²Δ𝑡^1/2+2𝑐²𝜆_{1[0,𝑇)𝕋}𝑢^𝜎(𝑡)^2𝛼Δ𝑡^1/2≤2𝜆_{1[0,𝑇)𝕋}𝑓^𝜎(𝑡)²Δ𝑡^1/2+2𝑐²𝑇^1−𝛼𝜆_{1[0,𝑇)𝕋}𝑢^𝜎(𝑡)²Δ𝑡^𝛼/2≤2𝜆_{1[0,𝑇)𝕋}𝑓^𝜎(𝑡)²Δ𝑡^1/2+2𝑐²𝑇^1−𝛼𝜆₁^1+𝛼‖𝑢‖^𝛼₁. (22) Let 𝜂₁∶=2/𝜆₁(∫_[0,𝑇)𝕋|𝑓^𝜎(𝑡)|²Δ𝑡)^1/2, and 𝜂₂∶=2𝑐²𝑇^1−𝛼/𝜆₁^1+𝛼. For any 𝑢∈𝐵(0,𝑅), ‖𝑆𝑢‖₁<𝑅,provided𝜂₁+𝜂₂𝑅^𝛼<𝑅. (23) Therefore, the above claim is true. Meanwhile, we also arrive at immediately 𝑆 is uniformly bounded on 𝐵(0,𝑅). On the other hand, we shall prove that 𝑆 is equicontinuous on 𝐵(0,𝑅). By (H3) and Hölders inequality, we have 𝑆𝑢₁−𝑆𝑢₂₁≤sup_{‖𝑣‖1=1[0,𝑇)𝕋}ℎ𝜎(𝑡),𝑢^𝜎₁(𝑡)−ℎ𝜎(𝑡),𝑢^𝜎₂(𝑡)𝑣^𝜎(𝑡)Δ𝑡≤sup_{‖𝑣‖1=1[0,𝑇)𝕋}𝑐𝑢^𝜎₁(𝑡)−𝑢^𝜎₂(𝑡)^𝛼𝑣^𝜎(𝑡)Δ𝑡≤𝑐sup_{‖𝑣‖1=1[0,𝑇)𝕋}𝑢^𝜎₁(𝑡)−𝑢^𝜎₂(𝑡)²Δ𝑡^𝛼/2×_[0,𝑇)𝕋𝑣^𝜎(𝑡)^2/(2−𝛼)Δ𝑡^(2−𝛼)/2≤𝑇^{(1−𝛼)/(2−𝛼)}𝑐sup_{‖𝑣‖1=1[0,𝑇)𝕋}𝑢^𝜎₁(𝑡)−𝑢^𝜎₂(𝑡)²Δ𝑡^𝛼/2×_[0,𝑇)𝕋𝑣^𝜎(𝑡)²Δ𝑡^1/(2−𝛼)≤𝑇^{(1−𝛼)/(2−𝛼)}𝑐𝜆₁^{1/(2−𝛼)+𝛼/2}𝑢₁−𝑢₂^𝛼₁. (24) We have from Arzelà-Ascoli theorem 𝑆 is compact on 𝐵(0,𝑅)⊂𝑊^1,2_Δ,𝑇.

Theorem 8. Assume that (H1)–(H3) hold; problem (3) has at least a weak solution.

Proof. We will utilize Leray-Schauder degree to prove the result. One is invited to verify that the existence of a solution of (3) is equivalent to the existence of a solution of the operator equation 𝑢=𝜆𝐴𝑢+𝑆𝑢,∀𝑢∈𝑊^1,2_Δ,𝑇, (25) where 𝐴 and 𝑆 are defined by (14) and (21), respectively. For the reason that 𝐴 is bounded, linear, and compact on 𝑊^1,2_Δ,𝑇, we easily see that Fréchet derivative (𝜆𝐴)^′ exists and 𝐼−(𝜆𝐴)^′ is continuously invertible by (H1). Consequently, Lemma 4 implies deg(𝐼−𝜆𝐴,𝐵(0,𝑅),0)≠0. (26) So, to complete the proof, we have to find an admissible homotopy connecting 𝐼−𝜆𝐴−𝑆(⋅) and 𝐼−𝜆𝐴. Define 𝐻(𝜏,𝑢)=𝑢−𝜆𝐴𝑢−𝜏𝑆𝑢,𝜏∈[0,1],𝑢∈𝑊^1,2_Δ,𝑇. (27) We shall prove that there exists 𝑅>0 such that for all 𝑢∈𝑊^1,2_Δ,𝑇, ‖𝑢‖₁=𝑅, and 𝜏∈[0,1], we obtain 𝐻(𝜏,𝑢)≠0. (28) If the claim is false, we can find sequences {𝑢_𝑛}⊂𝑊^1,2_Δ,𝑇 and {𝑡_𝑛}⊂[0,1] such that ‖𝑢_𝑛‖₁→∞ and 𝑢_𝑛−𝜆𝐴𝑢_𝑛−𝜏_𝑛𝑆𝑢_𝑛=0. (29) Set 𝑣_𝑛∶=𝑢_𝑛/‖𝑢_𝑛‖₁ and divide (29) by ‖𝑢_𝑛‖₁ to get 𝑣_𝑛−𝜆𝐴𝑣_𝑛−𝜏_𝑛𝑆𝑢_𝑛𝑢_𝑛1=0. (30) This is equivalent to _[0,𝑇)𝕋𝑣^Δ_𝑛(𝑡)𝑤^Δ(𝑡)Δ𝑡=𝜆_[0,𝑇)𝕋𝑣^𝜎_𝑛(𝑡)𝑤^𝜎(𝑡)Δ𝑡+𝜏_{𝑛[0,𝑇)𝕋}ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)𝑢_𝑛1𝑤^𝜎(𝑡)Δ𝑡, (31) for each 𝑤∈𝑊^1,2_Δ,𝑇. Now, passing to suitable subsequences, without loss of generality, we may assume that 𝜏_𝑛→𝜏∈[0,1] and 𝑣_𝑛→𝑣 in 𝑊^1,2_Δ,𝑇. Note that, similar with (22), we find _[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)𝑢_𝑛1𝑤^𝜎(𝑡)≤Δ𝑡_[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)𝑢_𝑛1𝑤^𝜎(𝑡≤𝜂)Δ𝑡₁+𝜂₂𝑢_𝑛^𝛼₁‖𝑤‖₁𝑢_𝑛1⟶0,𝑛⟶∞. (32) On the other hand, by the compactness of 𝐴, we have 𝐴𝑣_𝑛→𝐴𝑣 (see [15, Proposition 2.2.4(iii)]). In (30), let 𝑛→∞; we have 𝑣=𝜆𝐴𝑣, and 𝑣∈𝑊^1,2_Δ,𝑇 satisfies ‖𝑣‖₁=1. However, this contradicts our assumption 𝜆≠𝜆_𝑛, 𝑛=1,2,…. This implies that (28) holds. By Lemma 5 and (26), we have deg(𝐼−𝜆𝐴−𝑆,𝐵(0,𝑅),0)=deg(𝐼−𝜆𝐴,𝐵(0,𝑅),0)≠0. (33) Therefore, (3) has at least a weak solution.

3. Two Existence Theorems via Critical Point Theory

We still use the Sobolev’s space 𝑊^1,2_Δ,𝑇 defined by (4), which is equipped with the inner product (𝑢,𝑣)₂∶=_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡+_[0,𝑇)𝕋𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡, (34) and the corresponding norm ‖𝑢‖₂∶=_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡+_[0,𝑇)𝕋𝑢^Δ(𝑡)²Δ𝑡^1/2. (35) Now, we will establish the corresponding variational formulations for problem (3) as follows: 𝜑(𝑢)=12_[0,𝑇)𝕋𝑢^Δ(𝑡)²Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡−_[0,𝑇)𝕋𝐹𝜎(𝑡),𝑢^𝜎(𝑡)Δ𝑡, (36) where 𝐹(𝑡,𝜉)=∫^𝜉₀ℎ(𝑡,𝑠)d𝑠.

We now list our hypotheses for (3).(H4)𝐹(𝑡,𝑥) is Δ-measurable in 𝑡 for every 𝑥∈ℝ and continuously differentiable in 𝑥 for 𝑡∈[0,𝑇]_𝕋, and there exist 𝜖₁∈𝐶(ℝ⁺,ℝ⁺) and 𝜖₂∈𝐿¹_Δ([0,𝑇]_𝕋,ℝ⁺)|𝐹(𝑡,𝑥)|≤𝜖₁(|𝑥|)𝜖₂(𝑡),|ℎ(𝑡,𝑥)|≤𝜖₁(|𝑥|)𝜖₂(𝑡), (37) for all 𝑥∈ℝ and Δ-a.e. 𝑡∈[0,𝑇]_𝕋.(H5) There exist 𝑐₁>0 and 𝑝>0 such that |𝐹(𝑡,𝑥)|≤𝑐₁|𝑥|^𝑝,𝑡∈[0,𝑇]_𝕋,𝑢∈ℝ. (38)(H6) There exist 𝜀₁,𝜀₂>0 such that ℎ(𝑡,𝑥)𝑥−2𝐹(𝑡,𝑥)≥𝜀₁|𝑥|−𝜀₂,𝑡∈[0,𝑇]_𝕋,𝑢∈ℝ. (39)(H7)  lim_|𝑥|→∞(𝐹(𝑡,𝑥)/|𝑥|²)=+∞ uniformly for 𝑡∈[0,𝑇]_𝕋.(H8)  𝐹(𝑡,−𝑥)=𝐹(𝑡,𝑥), 𝑡∈[0,𝑇]_𝕋, 𝑢∈ℝ.

Remark 9. Let 𝐹(𝑡,𝑥)=^𝑛_𝑖=11(2𝑖)²𝑥^2𝑖, (40) where 𝑛>1 is a fixed positive integer. Then ℎ(𝑡,𝑥)=^𝑛_𝑖=11𝑥2𝑖^2𝑖−1. (41) Direct computation shows lim_|𝑥|→∞𝐹(𝑡,𝑥)|𝑥|²=lim_|𝑥|→∞∑^𝑛_𝑖=11/(2𝑖)²𝑥^2𝑖|𝑥|²=+∞,lim_|𝑥|→∞ℎ(𝑡,𝑥)𝑥−2𝐹(𝑡,𝑥)|𝑥|=lim_|𝑥|→∞∑^𝑛_𝑖=11/2𝑖−1/2𝑖²𝑥^2𝑖|𝑥|=+∞. (42) Clearly, (H4)–(H8) hold.
By (H4), we find 𝜑∈𝐶¹(𝑊^1,2_Δ,𝑇,ℝ) (see [11, Theorem 2.27]), and for 𝑢,𝑣∈𝑊^1,2_Δ,𝑇, 𝜑^′(𝑢)𝑣=_[0,𝑇)𝕋𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡−_[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡. (43) Clearly, the existence of weak solutions for (3) is equivalent to the existence of critical points for 𝜑. In what follows, we take 𝑎∶=min{1,−𝜆}>0, 𝑏∶=max{1,−𝜆}>0.

Lemma 10 (see [17, Theorem 1.2]). Suppose 𝑋 is a reflexive Banach space with norm ‖⋅‖, and 𝜑∶𝑋→ℝ∪{+∞} is coercive and weak (sequentially) lower semicontinuous; that is, the following conditions are fulfilled as follows:(1)  𝜑(𝑢)→∞ as ‖𝑢‖→∞, 𝑢∈𝑋.(2)  For any 𝑢∈𝑋, any sequence {𝑢_𝑚}⊂𝑋 such that 𝑢_𝑚⇀𝑢, there holds 𝜑(𝑢)≤liminf_𝑚→∞𝜑𝑢_𝑚. (44) Then 𝜑 is bounded from below and attains its infimum on 𝑋.

Theorem 11. If (H4) and (H5) with 𝑝∈(0,2) hold, (3) has at least a weak solution.

Proof. Our working space 𝑊^1,2_Δ,𝑇 is a Hilbert space, so it is reflexive. By Lemma 2.1 and Theorem 3.3 in [8], we see 𝜑 is weakly lower semicontinuous on 𝑊^1,2_Δ,𝑇. On the other hand, by (H5), we have 𝜑(𝑢)≥𝑎2‖𝑢‖²₂−_[0,𝑇)𝕋𝑐₁𝑢^𝜎(𝑡)^𝑝≥Δ𝑡𝑎2‖𝑢‖²₂−𝑐₁𝑇^(2−𝑝)/2‖𝑢‖^𝑝₂, (45) and thus 𝜑(𝑢)→∞ as ‖𝑢‖₂→∞. Lemma 10 implies 𝜑 can attain its infimum in 𝑋; that is, (3) has at least a weak solution. This completes the proof.

If 𝑋 is a Hilbert space, there exist (see [18]) {𝑒_𝑛}^∞_𝑛=1⊂𝑋 and {𝑓_𝑛}^∞_𝑛=1⊂𝑋^∗ such that 𝑓_𝑛(𝑒_𝑚)=𝛿_𝑛,𝑚, 𝑋=span{𝑒_𝑛∶𝑛=1,2,…} and 𝑋^∗=span^𝑊∗{𝑓_𝑛∶𝑛=1,2,…}. For 𝑗,𝑘∈ℕ, denote 𝑋_𝑗∶=span{𝑒_𝑗}, 𝑌_𝑘∶=⊕^𝑘_𝑗=1𝑋_𝑗, and 𝑍_𝑘∶=⊕^∞_𝑗=𝑘+1𝑋_𝑗. Clearly, 𝑋=⊕_𝑗∈ℕ𝑋_𝑗 with dim𝑋_𝑗<∞ for all 𝑗∈ℕ. Since 𝑊^1,2_Δ,𝑇 is a Hilbert space, we can choose an orthonormal basis {𝑒_𝑛} such that 𝑊^1,2_Δ,𝑇=span𝑒_𝑛∶𝑛=1,2…,𝑌_𝑘∶=^𝑘_𝑗=1𝑋_𝑗,𝑍_𝑘∶=^∞_𝑗=𝑘+1𝑋_𝑗,where𝑋_𝑗∶=span𝑒_𝑗. (46)

Definition 12 (see [19, Definition 1.1]). Assume that 𝑋 is a Banach space with norm ‖⋅‖; we say that 𝜑∈𝐶¹(𝑋,ℝ) satisfies Cerami condition (C) if for all 𝑑∈ℝ,(i)any bounded sequence {𝑢_𝑛}⊂𝑋 satisfying 𝜑(𝑢_𝑛)→𝑑, 𝜑^′(𝑢_𝑛)→0 possesses a convergent subsequence;(ii)there exist 𝛿,𝜉,𝜌>0 such that for any 𝑢∈𝜑⁻¹([𝑑−𝛿,𝑑+𝛿]) with ‖𝑢‖≥𝜉, ‖𝜑^′(𝑢)‖⋅‖𝑢‖≥𝜌. Denote 𝑆_𝜌∶={𝑢∈𝑋∶‖𝑢‖=𝜌}. We will introduce the following Fountain theorem under condition (C).

Lemma 13 (see [19, Proposition 1.2]). Assume that 𝜑∈𝐶¹(𝑋,ℝ) satisfies condition (C), and 𝜑(−𝑢)=𝜑(𝑢). For each 𝑘∈ℕ, there exists 𝜌_𝑘>𝑟_𝑘>0 such that(i)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘∩𝑆𝑟𝑘}𝜑(𝑢)→+∞, 𝑘→∞;(ii)𝑎_𝑘∶=max_{𝑢∈𝑌𝑘∩𝑆𝜌𝑘}𝜑(𝑢)≤0.
Then 𝜑 has a sequence of critical points 𝑢_𝑛, such that 𝜑(𝑢_𝑛)→+∞ as 𝑛→∞.

Lemma 14. Suppose that (H4), (H5), and (H6) hold; then 𝜑 satisfies the condition (C).

Proof. For all 𝑑∈ℝ, we assume that {𝑢_𝑛}∈𝑊^1,2_Δ,𝑇 is bounded and 𝜑𝑢_𝑛⟶𝑑,𝜑^′𝑢_𝑛⟶0,𝑛⟶∞. (47) Going, if necessary, to a subsequence, we can assume that 𝑢_𝑛⇀𝑢 in 𝑊^1,2_Δ,𝑇; then 𝜑^′𝑢_𝑛−𝜑^′(𝑢)𝑢_𝑛=−𝑢_[0,𝑇)𝕋𝑢^Δ_𝑛(𝑡)−𝑢^Δ(𝑡)²Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎_𝑛(𝑡)−𝑢^𝜎(𝑡)²Δ𝑡−_[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)−ℎ𝜎(𝑡),𝑢^𝜎(𝑡)×𝑢^𝜎_𝑛(𝑡)−𝑢^𝜎(𝑡)Δ𝑡. (48) Lemma 1 leads to _[0,𝑇)𝕋ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)−ℎ𝜎(𝑡),𝑢^𝜎(𝑡)×𝑢^𝜎_𝑛(𝑡)−𝑢^𝜎(𝑡)Δ𝑡⟶0,𝑛⟶∞. (49) It follows that 𝑢_𝑛⇀𝑢 in 𝑊^1,2_Δ,𝑇 and (𝜑^′(𝑢_𝑛)−𝜑^′(𝑢))(𝑢_𝑛−𝑢)→0, we see _[0,𝑇)𝕋𝑢^Δ_𝑛(𝑡)−𝑢^Δ(𝑡)²Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎_𝑛(𝑡)−𝑢^𝜎(𝑡)²Δ𝑡⟶0,𝑛⟶∞. (50)
Clearly, it is equivalent to ‖⋅‖₂, and then we have 𝑢_𝑛−𝑢₂⟶0,𝑛⟶∞. (51) Hence, condition (i) of Definition 12 holds. Next, we prove condition (ii) of Definition 12, suppose the contrary, there exists a sequence {𝑢_𝑛}⊂𝑊^1,2_Δ,𝑇 such that 𝜑𝑢_𝑛⟶𝑑,𝜑^′𝑢_𝑛2⋅𝑢_𝑛2𝑢⟶0,𝑛⟶∞,_𝑛2⟶∞,𝑛⟶∞. (52)(53) By (52), there exists a constant 𝜀₃>0 such that 𝜑𝑢_𝑛−12𝜑^′𝑢_𝑛𝑢_𝑛≤𝑐₃. (54) On the other hand, (H6) implies 𝜑𝑢_𝑛−12𝜑^′𝑢_𝑛𝑢_𝑛=_[0,𝑇)𝕋12ℎ𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)𝑢^𝜎_𝑛(𝑡)−𝐹𝜎(𝑡),𝑢^𝜎_𝑛(𝑡)≥1Δ𝑡2𝜀_{1[0,𝑇)𝕋}𝑢^𝜎_𝑛(𝑡1)Δ𝑡−2𝜀₂𝑇. (55) This, together with (54), leads to there is a constant 𝜀₄>0 such that max_{𝑡∈[0,𝑇]𝕋}|𝑢^𝜎_𝑛(𝑡)|≤𝜀₄. By (H5), similar with (45), we have 𝜑𝑢_𝑛≥𝑎2𝑢_𝑛²₂−_[0,𝑇)𝕋𝑐₁𝑢^𝜎_𝑛(𝑡)^𝑝Δ𝑡≥𝑎2𝑢_𝑛²₂−𝑐₁𝑇𝜀^𝑝₄, (56) and thus 𝜑(𝑢_𝑛)→∞ if (53) holds, which contradicts 𝜑(𝑢_𝑛)→𝑑 in (52). This proves that 𝜑 satisfies condition (C).

Theorem 15. Under assumptions (H4)–(H8) with 𝑝>2 in (H5), problem (3) has infinitely many solutions.

Proof. (H8) and Lemma 14 enable us to obtain that 𝜑(𝑢)=𝜑(−𝑢) and 𝜑 satisfies the condition (C). For any 𝑢∈𝑌_𝑘, let ‖𝑢‖_∗∶=_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡^1/2, (57) and it is easy to verify that ‖⋅‖ defined by (57) is a norm of 𝑌_𝑘. Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant 𝜀₅ such that 𝜀₅‖𝑢‖₂≤‖𝑢‖_∗. In view of (H7), there exist 𝜀₆>𝑏/2𝜀²₅ and 𝜚>0 such that 𝐹𝜎(𝑡),𝑢^𝜎(𝑡)≥𝜀₆𝑢^𝜎(𝑡)²−𝜚,𝑡∈[0,𝑇]_𝕋,𝑢∈ℝ. (58) This implies that 𝜑(𝑢)=12_[0,𝑇)𝕋𝑢^Δ(𝑡)²Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡−_[0,𝑇)𝕋𝐹𝜎(𝑡),𝑢^𝜎(𝑡)≤Δ𝑡𝑏2‖𝑢‖²₂−𝜀₆𝜀²₅‖𝑢‖²₂+𝜚𝑇. (59) Since 𝑏/2−𝜀₆𝜀²₅<0; then there exists positive constant 𝑑_𝑘 such that 𝜑(𝑢)≤0,foreach𝑢∈𝑌_𝑘,‖𝑢‖≥𝑑_𝑘. (60) Let 𝛽_𝑘∶=sup_{𝑢∈𝑍𝑘,‖𝑢‖2=1}|𝑢(𝑡)|, 𝑡∈[0,𝑇]_𝕋. Then by Lemma 3.8 of [20] and Lemma 1 we obtain 𝛽_𝑘→0, as 𝑘→∞. For any 𝑢∈𝑍_𝑘, note that 𝑝∈(1,2), in view of (H5), we find that 𝜑(𝑢)=12_[0,𝑇)𝕋𝑢^Δ(𝑡)²Δ𝑡−𝜆_[0,𝑇)𝕋𝑢^𝜎(𝑡)²Δ𝑡−_[0,𝑇)𝕋𝐹𝜎(𝑡),𝑢^𝜎(𝑡)≥Δ𝑡𝑎2‖𝑢‖²₂−𝑐_{1[0,𝑇)𝕋}𝑢^𝜎(𝑡)^𝑝≥Δ𝑡𝑎2‖𝑢‖²₂−𝑐₁𝑇‖𝑢‖^𝑝_∞≥𝑎2‖𝑢‖²₂−𝑐₁𝑇𝛽^𝑝_𝑘‖𝑢‖^𝑝₂. (61) Choosing ‖𝑢‖₂=𝑟_𝑘∶=1/𝛽_𝑘, then 𝑟_𝑘→∞ as 𝑘→∞, then we have; 𝜑(𝑢)≥𝑎2𝑟²_𝑘−𝑐₁𝑇⟶∞,as𝑘⟶∞. (62) Hence, 𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝑟𝑘}𝜑(𝑢)→∞ as 𝑘→∞. Combining this and (60), we can take 𝜌_𝑘∶=max{𝑑_𝑘,𝑟_𝑘+1}, and thus 𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝜑(𝑢)≤0. Up until now, we have proved that the functional 𝜑 satisfies all the conditions of Lemma 13; then 𝜑 has infinitely many solutions.

Acknowledgment

This research is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J13LI51).