Multiple Solutions for a Class of Fractional Boundary Value Problems

Abstract

We study the multiplicity of solutions for the following fractional boundary value problem: (𝑑/𝑑𝑡)((1/2)₀𝐷_𝑡^−𝛽(𝑢^′(𝑡))+(1/2)₀𝐷_𝑇^−𝛽(𝑢^′(𝑡)))+𝜆∇𝐹(𝑡,𝑢(𝑡))=0,a.e.𝑡∈[0,𝑇],𝑢(0)=𝑢(𝑇)=0, where ₀𝐷_𝑡^−𝛽 and ₀𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 0≤𝛽<1, respectively, 𝜆>0 is a real number, 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ is a given function, and ∇𝐹(𝑡,𝑥) is the gradient of 𝐹 at 𝑥. The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.

1. Introduction

In this paper, we consider the fractional boundary value problem of the following form: 𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12⁰𝐷_𝑇^−𝛽𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇)+𝜆∇𝐹𝑡,𝑢))=0,a.a.𝑡∈0,𝑇=𝑢=0, (𝑃) where ₀𝐷_𝑡^−𝛽 and ₀𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 0≤𝛽<1, respectively, 𝜆>0 is a real number, 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ is a given function, and ∇𝐹(𝑡,𝑥) is the gradient of 𝐹 at 𝑥.

In particular, if 𝜆=1, the problem 𝑃₁ reduces to the standard second-order boundary value problem of the following form: 𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12⁰𝐷_𝑇^−𝛽𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇)+∇𝐹𝑡,𝑢))=0,a.a.𝑡∈0,𝑇=𝑢=0. 𝑃₁

Fractional calculus and fractional differential equations can find many applications in various fields of physical science such as viscoelasticity, diffusion, control, relaxation processes, and modeling phenomena in engineering, see [1–12]. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by use of techniques of nonlinear analysis, such as fixed-point theory (including Leray-Schauder nonlinear alternative) (see [13, 14]), topological degree theory (including coincidence degree theory) (see [15, 16]), and comparison method (including upper and lower solutions methods and monotone iterative method) (see [17, 18]). However, it seems that the popular methods mentioned above are not appropriate for discussing (𝑃) and 𝑃₁, as the equivalent integral equation is not easy to be obtained.

In the past, there were investigations of the eigenvalue problems for fractional differential equations. For more detailed information on this topic, we refer to Zhang et al. [19–21], Wang et al. [22, 23], and Jiang et al. [24].

Recently, there are many papers dealing with the existence of solutions for problem 𝑃₁. In [25], Jiao and Zhou obtained the existence of solutions for 𝑃₁ by mountain pass theorem under the Ambrosetti-Rabinowitz condition. Chen and Tang [26] studied the existence and multiplicity of solutions for the system 𝑃₁ when the nonlinearity 𝐹(𝑡,⋅) is superquadratic, asymptotically quadratic, and subquadratic, respectively.

But so far, few papers discuss the two solutions of the system (𝑃) via critical point theory. The aim of the present paper is to study the existence of at least two solutions for the system (𝑃) as the parameter 𝜆>𝜆₀ for some constant 𝜆₀.

The paper is organized as follows. We first introduce some basic preliminary results and a well-known lemma in Section 2, including the fractional derivative space 𝐸^𝛼, where 𝛼∈(1/2,1]. In Section 3, we give the main result and its proof. In Section 4, we give the summary of this paper.

2. Preliminary

In this section, we recall some related preliminaries and display the variational setting which has been established for our problem.

Definition 2.1 (see [8]). Let 𝑓(𝑡) be a function defined on [𝑎,𝑏] and 𝜏>0. The left and right Riemann-Liouville fractional integrals of order 𝜏 for function 𝑓(𝑡) denoted by _𝑎𝐷_𝑡^−𝜏𝑓(𝑡) and _𝑡𝐷_𝑏^−𝜏𝑓(𝑡), respectively, are defined by _𝑎𝐷_𝑡^−𝜏𝑓(𝑡)=1Γ(𝜏)^𝑡_𝑎(𝑡−𝑠)^𝜏−1𝑓(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏_𝑡],𝐷_𝑏^−𝜏𝑓(𝑡)=1Γ(𝜏)^𝑏_𝑡(𝑡−𝑠)^𝜏−1𝑓(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏], (2.1) provided the right-hand sides are pointwise defined on [𝑎,𝑏], where Γ is the gamma function.

Definition 2.2 (see [8]). Let 𝑓(𝑡) be a function defined on [𝑎,𝑏]. The left and right Riemann-Liouville fractional derivatives of order 𝜏 for function 𝑓(𝑡) denoted by _𝑎𝐷^𝜏_𝑏𝑓(𝑡) and _𝑡𝐷^𝜏_𝑏𝑓(𝑡), respectively, are defined by _𝑎𝐷^𝜏_𝑡𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝑎𝐷_𝑡^𝜏−𝑛𝑓(𝑡)=1Γ()𝑑𝑛−𝜏𝑑𝑡^𝑛𝑛𝑡_𝑎(𝑡−𝑠)^{𝑛−𝜏−1}𝑓(𝑠)𝑑𝑠_𝑡,𝐷^𝜏_𝑏𝑓(𝑡)=(−1)^𝑛𝑑^𝑛𝑑𝑡^𝑛𝑡𝐷_𝑏^𝜏−𝑛𝑓(𝑡)=1Γ()𝑑𝑛−𝜏𝑑𝑡^𝑛𝑛𝑏_𝑡(𝑡−𝑠)^{𝑛−𝜏−1}𝑓(𝑠)𝑑𝑠, (2.2) where 𝑡∈[𝑎,𝑏], 𝑛−1≤𝜏<𝑛, and 𝑛∈ℕ.

The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belong-ing to the space of absolutely continuous functions, which we denote by 𝐴𝐶([𝑎,𝑏],ℝ^𝑁). 𝐴𝐶^𝑘([𝑎,𝑏],ℝ^𝑁)(𝑘=1,2,…) is the space of functions 𝑓 such that 𝑓∈𝐶^𝑘([𝑎,𝑏],ℝ^𝑁). In particular, 𝐴𝐶([𝑎,𝑏],ℝ^𝑁)=𝐴𝐶¹([𝑎,𝑏],ℝ^𝑁).

Definition 2.3 (see [8]). Let 𝜏≥0 and 𝑛∈ℕ. If 𝜏∈[𝑛−1,𝑛) and 𝑓(𝑡)∈𝐴𝐶^𝑛([𝑎,𝑏],ℝ^𝑁), then the left and right Caputo fractional derivatives of order 𝜏 for function 𝑓(𝑡) denoted by ^𝑐_𝑎𝐷^𝜏_𝑡𝑓(𝑡) and ^𝑐_𝑡𝐷^𝜏_𝑏𝑓(𝑡), respectively, which exist a.e. on [𝑎,𝑏]. ^𝑐_𝑎𝐷^𝜏_𝑡𝑓(𝑡) and ^𝑐_𝑡𝐷^𝜏_𝑏𝑓(𝑡) are represented by ^𝑐_𝑎𝐷^𝜏_𝑡𝑓(𝑡)=_𝑎𝐷_𝑡^𝜏−𝑛𝑓^(𝑛)(𝑡)=1Γ(𝑛−𝜏)^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝜏−1}𝑓^(𝑛)(𝑠)𝑑𝑠_𝑐𝑡,𝐷^𝜏_𝑏𝑓(𝑡)=(−1)^𝑛_𝑡𝐷_𝑏^𝜏−𝑛𝑓^(𝑛)(𝑡)=1Γ(𝑛−𝜏)^𝑏_𝑡(𝑡−𝑠)^{𝑛−𝜏−1}𝑓^(𝑛)(𝑠)𝑑𝑠, (2.3) respectively, where 𝑡∈[𝑎,𝑏].

Definition 2.4 (see [25]). Define 0<𝛼≤1 and 1<𝑝<∞. The fractional derivative space 𝐸₀^𝛼,𝑝 is defined by the closure of 𝐶^∞₀([0,𝑇],ℝ^𝑁) with respect to the norm ‖𝑢‖_𝛼,𝑝=^𝑇₀|𝑢(𝑡)|^𝑝𝑑𝑡+^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡)^𝑝𝑑𝑡^1/𝑝,∀𝑢∈𝐸₀^𝛼,𝑝, (2.4) where 𝐶^∞₀([0,𝑇],ℝ^𝑁) denotes the set of all functions 𝑢∈𝐶^∞([0,𝑇],ℝ^𝑁) with 𝑢(0)=𝑢(𝑇)=0. It is obvious that the fractional derivative space 𝐸₀^𝛼,𝑝 is the space of functions 𝑢∈𝐿^𝑝([0,𝑇],ℝ^𝑁) having an 𝛼-order Caputo fractional derivative ^𝑐₀𝐷^𝛼_𝑡𝑢∈𝐿^𝑝([0,𝑇],ℝ^𝑁) and 𝑢(0)=𝑢(𝑇)=0.

Proposition 2.5 (see [25]). Let 0<𝛼≤1 and 1<𝑝<∞. The fractional derivative space 𝐸₀^𝛼,𝑝 is a reflexive and separable space.

Proposition 2.6 (see [25]). Let 0<𝛼≤1 and 1<𝑝<∞. For all 𝑢∈𝐸₀^𝛼,𝑝, one has ‖𝑢‖_𝐿^𝑝≤𝑇^𝛼Γ(𝛼+1)^𝑐0𝐷^𝛼_𝑡𝑢_𝐿𝑝. (2.5) Moreover, if 𝛼>1/𝑝 and 1/𝑝+1/𝑞=1, then ‖𝑢‖_∞≤𝑇^{(𝛼−1)/𝑝}Γ(𝛼)(𝛼−1)𝑞+1^1/𝑞𝑐0𝐷^𝛼_𝑡𝑢_𝐿𝑝. (2.6) According to (2.6), one can consider 𝐸₀^𝛼,𝑝 with respect to the norm ‖𝑢‖_𝛼,𝑝=^𝑐₀𝐷^𝛼_𝑡𝑢_𝐿𝑝=^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢^𝑝𝑑𝑡^1/𝑝. (2.7)

Proposition 2.7 (see [25]). Define 0<𝛼≤1 and 1<𝑝<∞. Assume that 𝛼>1/𝑝, and the sequence 𝑢_𝑘 converges weakly to 𝑢∈𝐸₀^𝛼,𝑝, that is, 𝑢_𝑘⇀𝑢. Then 𝑢_𝑘→𝑢 in 𝐶([0,𝑇],ℝ^𝑁), that is, ‖𝑢_𝑘−𝑢‖_∞→0, as 𝑘→∞.

Making use of Definition 2.3, for any 𝑢∈𝐴𝐶([0,𝑇],ℝ^𝑁), problem (𝑃) is equivalent to the following problem: 𝑑1𝑑𝑡2⁰𝐷_𝑡^𝛼−1𝑐₀𝐷^𝛼_𝑡𝑢(𝑡)−12^𝑡𝐷_𝑇^𝛼−1𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)((𝑡[],𝑢(0)(𝑇)+𝜆∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇=𝑢=0, 𝑃₂ where 𝛼=1−𝛽/2∈(1/2,1].

In the following, we will treat problem 𝑃₂ in the Hilbert space 𝐸^𝛼=𝐸₀^𝛼,2 with the corresponding norm ‖𝑢‖_𝛼=‖𝑢‖_𝛼,2. It follows from [25, Theorem 4.1] that the functional 𝜑 given by 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡 (2.8) is continuously differentiable on 𝐸^𝛼. Moreover, for 𝑢,𝑣∈𝐸^𝛼, we have 𝜑^′(𝑢),𝑣=−^𝑇₀12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑣(𝑡)+^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑣(𝑡)𝑑𝑡−𝜆^𝑇₀(∇𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡)]𝑑𝑡. (2.9)

Definition 2.8 (see [25]). A function 𝑢∈𝐴𝐶([0,𝑇],ℝ^𝑁) is called a solution of 𝑃₂ if
(i) 𝐷^𝛼(𝑢(𝑡)) is derivative for a.e. 𝑡∈[0,𝑇],
(ii) 𝑢 satisfies 𝑃₂, where 𝐷^𝛼(𝑢(𝑡))∶=(1/2)₀𝐷_𝑡^𝛼−1(^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡))−(1/2)_𝑡𝐷_𝑇^𝛼−1(^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)).

Proposition 2.9 (see [25]). If 1/2<𝛼≤1, then for any 𝑢∈𝐸^𝛼, one has |cos(𝜋𝛼)|‖𝑢‖²_𝛼≤−^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)1𝑑𝑡≤|(‖𝑢‖cos𝜋𝛼)|^2𝛼. (2.10)

Proposition 2.10 (see [25]). Let 1/2<𝛼≤1 be satisfied. If 𝑢∈𝐸^𝛼, then the functional 𝐽∶𝐸^𝛼→ℝ denoted by 𝐽(𝑢)1=−2^𝑇₀^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡 (2.11) is convex and continuous on 𝐸^𝛼.

In order to prove the existence of two solutions for problem (𝑃), firstly, we recall some well-known results. Their proofs can be found in many books. Please refer to the references and its references therein.

Lemma 2.11 (see [27]). If 𝑋 is a Banach space, 𝜑∈𝐶¹(𝑋,ℝ), 𝑒∈𝑋, and 𝑟>0, such that ‖𝑒‖>𝑟 and 𝑏∶=inf_{‖𝑢‖=𝑟}𝜑(𝑢)>𝜑(0)≥𝜑(𝑒), (2.12) and if 𝜑 satisfies the PS condition, with 𝑐∶=inf_𝛾∈Γmax_[]𝑡∈0,1𝜑𝛾(𝑡),Γ∶=𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒, (2.13) then 𝑐 is a critical value of 𝜑.

3. The Main Result and Proof of the Theorem

In this part, we will prove that for (𝑃) there also exist two solutions for the general case.

Our hypotheses on nonsmooth potential 𝐹(𝑡,𝑥) are as follows.

H(F)₁: 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ is a function such that 𝐹(𝑡,0)=0 a.e. on [0,𝑇] and satisfies the following facts:(i)for all 𝑥∈ℝ^𝑁, 𝑡↦𝐹(𝑡,𝑥) is measurable,(ii)for a.a. 𝑡∈[0,𝑇], 𝑥↦𝐹(𝑡,𝑥) is continuously differentiable,(iii) there exist 𝑐∈𝐶([0,𝑇],ℝ) and 0<𝛼₀<2, such that |∇𝐹(𝑡,𝑥)|,|𝐹(𝑡,𝑥)|≤𝑐(𝑡)1+|𝑥|^𝛼0,fora.a.𝑡∈[0,𝑇],all𝑥∈ℝ^𝑁, (3.1)(iv)there exist 𝛾>2 and 𝜇∈𝐿^∞([0,𝑇]), such that limsup_|𝑥|→0(∇𝐹(𝑡,𝑥),𝑥)|𝑥|^𝛾<𝜇(𝑡),uniformlyfora.a.𝑡∈[0,𝑇], (3.2)(v)there exist 𝜉₀∈ℝ^𝑁, 𝑡₀∈(0,𝑇), and 𝑟₀>0, such that 𝐹𝑡,𝜉₀>𝛿₀>0,a.e.𝑡∈𝐵_𝑟0𝑡₀, (3.3) where 𝐵_𝑟0(𝑡₀)∶={𝑡∈[0,𝑇]∶|𝑡−𝑡₀|≤𝑟₀}⊂[0,𝑇].

Remark 3.1. It is easy to verify that 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ satisfies the whole assumption in [25, Theorem 4.1]. So, 𝑢∈𝐸^𝛼 is a solution of the corresponding Euler equation 𝜑^′(𝑢)=0, then 𝑢 is a solution of problem 𝑃₂ which, of course, corresponds to the solution of problem (𝑃). For great details, please see [25, Theorem 4.2].

Theorem 3.2. Suppose that H(F)₁ holds. Then there exists 𝜆₀>0 such that for each 𝜆>𝜆₀, the problem 𝑃₂ has at least two nontrivial solutions, which correspond to the two solutions of problem (𝑃).

Proof. The proof is divided into four steps as follows.
Step 1. We will show that 𝜑 is coercive in this step.
Firstly, by H(F)₁(iii), (2.6), and (2.10), we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹((𝑡≥|(𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−𝑐₀𝜆𝑇−𝑐₀𝜆^𝑇₀|𝑢(𝑡)|^𝛼0≥|(𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−𝑐₀𝜆𝑇−𝑐₀𝑇𝜆𝑇^(𝛼−1)/2√2𝛼−1Γ(𝛼)^𝛼0‖𝑢‖^𝛼0_𝛼→∞,as‖𝑢‖_𝛼→∞, (3.4) where 𝑐₀=max_{𝑡∈[0,𝑇]}|𝑐(𝑡)|.
Step 2. We will show that the 𝜑 is weakly lower semicontinuous.
Let 𝑢_𝑛⇀𝑢 weakly in 𝐸^𝛼, and by Proposition 2.7, we obtain the following results: 𝐸^𝛼↪𝐶[0,𝑇],ℝ^𝑁,𝑢_𝑛(𝑡)(𝑡)[],𝐹→𝑢fora.e.𝑡∈0,𝑇𝑡,𝑢_𝑛(𝑡)→𝐹(𝑡,𝑢(𝑡))fora.e.𝑡∈[0,𝑇]. (3.5)
By Fatou's lemma, limsup_𝑛→∞^𝑇₀𝐹𝑡,𝑢_𝑛(𝑡)𝑑𝑡≤^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡. (3.6) On the other hand, by Proposition 2.10, we have lim_𝑛→∞𝐽(𝑢_𝑛)=𝐽(𝑢), that is, lim_𝑛→∞^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢_𝑛(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)𝑑𝑡=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡. (3.7)
Thus, liminf_𝑛→∞𝜑𝑢_𝑛=liminf_𝑛→∞^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢_𝑛(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)𝑑𝑡−limsup_𝑛→∞𝜆^𝑇₀𝐹𝑡,𝑢_𝑛(𝑡)≥𝑑𝑡_𝑇0−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡=𝜑(𝑢). (3.8)
Hence, by the Weierstrass theorem, we deduce that there exists a global minimizer 𝑢₀∈𝐸^𝛼 such that 𝜑𝑢₀=min_{𝑢∈𝐸𝛼}𝜑(𝑢). (3.9)
Step 3. We will show that there exists 𝜆₀>0 such that for each 𝜆>𝜆₀, 𝜑(𝑢₀)<0.
By the condition by H(F)₁ (v), there exists 𝜉₀∈ℝ^𝑁 such that 𝐹(𝑡,𝜉₀)>𝛿₀>0, a.e. 𝑡∈𝐵_𝑟0(𝑡₀). It is clear that 0<𝑀₁∶=max_{|𝑥|≤|𝜉0|}𝑐₀+𝑐₀|𝑥|^𝛼0<+∞. (3.10)
Now we denote 𝜅₀=𝑀₁𝛿₀+𝑀₁,𝜆₀=max_{𝜅∈[𝜅1,𝜅2]}𝜉₀²𝑇^3−2𝛼2Γ²(1−𝛼)𝑟³₀(1−𝜅)²(1−𝛼)²(3−2𝛼)𝛿₀𝜅−𝑀₁+𝜅𝑀₁|(,cos𝜋𝛼)| (3.11) where 𝜅₀<𝜅₁<𝜅₂<1, and 𝛿₀ is given in the condition H(F)₁ (v). A simple calculation shows that the function 𝜅↦𝛿₀𝜅−𝑀₁+𝜅𝑀₁ is positive whenever 𝜅>𝜅₀ and 𝛿₀𝜅₀−𝑀₁+𝜅₀𝑀₁=0. Thus, 𝜆₀ is well defined and 𝜆₀>0.
We will show that for each 𝜆>𝜆₀, the problem 𝑃₂ has two nontrivial solutions. In order to do this, for 𝑡∈[𝑡₁,𝑡₂], let us define 𝜂_𝜅(𝑡)=0if𝑡∈[0,𝑇]/𝐵_𝑟0𝑡₀,𝜉₀if𝑡∈𝐵_𝜅𝑟0𝑡₀,𝜉₀𝑟₀()𝑟1−𝜅⁰−𝑡−𝑡₀if𝑡∈𝐵_𝑟0𝑡₀/𝐵_𝜅𝑟0𝑡₀. (3.12)
Then |𝜂^′_𝜅(𝑡)|≤|𝜉₀|/𝑟₀(1−𝜅) and ^𝑐₀𝐷^𝛼_𝑡𝜂_𝜅(𝑡≤1)Γ(1−𝛼)^𝑡₀(𝑡−𝑠)^−𝛼𝜂^′_𝜅(𝑠1)𝑑𝑠≤Γ()𝜉1−𝛼𝑟₀(⁰)𝑡1−𝜅^1−𝛼.1−𝛼 (3.13) Hence, 𝜂_𝜄²_𝛼=^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝜂_𝜅(𝑡)²𝜉𝑑𝑡≤⁰²𝑇^3−2𝛼Γ²(1−𝛼)𝑟²₀(1−𝜅)²(1−𝛼)²().3−2𝛼 (3.14) By H(F)₁ (iii) and (v), we have ^𝑇₀𝐹𝑡,𝜂_𝜅(𝑡)𝑑𝑡=_{𝐵0𝜅𝑟(𝑡0)}𝐹𝑡,𝜂_𝜅(𝑡)𝑑𝑡+_{𝐵𝑟0(𝑡0)⧵𝐵0𝜅𝑟(𝑡0)}𝐹𝑡,𝜂_𝜅(𝑡)𝑑𝑡≥2𝜅𝑟₀𝛿₀−𝑀₁2𝑟₀−2𝜅𝑟₀=2𝑟₀𝛿₀𝜅−𝑀₁+𝜅𝑀₁. (3.15)
For 𝜅∈[𝜅₁,𝜅₂], by Proposition 2.9, we have 𝜑𝜂_𝜅=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝜂_𝜅(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝜂_𝜅(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹𝑡,𝜂_𝜅(𝑡)≤1𝑑𝑡|(𝜂cos𝜋𝛼)|^𝜅2𝛼−2𝜆𝑟₀𝛿₀𝜅−𝑀₁+𝜅𝑀₁≤1|(𝜉cos𝜋𝛼)|⁰²𝑇^3−2𝛼Γ²(1−𝛼)𝑟²₀(1−𝜅)²(1−𝛼)²(3−2𝛼)−2𝜆𝑟₀𝛿₀𝜅−𝑀₁+𝜅𝑀₁, (3.16) so that 𝜑(𝜂_𝑡)<0 whenever 𝜆>𝜆₀.
Step 4. We will check the PS condition in the following.
Suppose that {𝑢_𝑛}_𝑛≥1⊆𝐸^𝛼 such that 𝜑(𝑢_𝑛)→𝑐 and ‖𝜑^′(𝑢_𝑛)‖_𝛼→0.
Since 𝜑 is coercive and {𝑢_𝑛}_𝑛≥1 is bounded in 𝐸^𝛼 and passed to a subsequence, which still denote {𝑢_𝑛}_𝑛≥1, we may assume that there exists 𝑢∈𝐸^𝛼, such that 𝑢_𝑛⇀𝑢 weakly in 𝐸^𝛼; thus, we have 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛−𝑢=𝜑^′𝑢_𝑛,𝑢_𝑛−𝑢−𝜑^′(𝑢),𝑢_𝑛≤𝜑−𝑢_′𝑢_𝑛𝑢_𝑛𝛼−𝜑^′(𝑢),𝑢_𝑛−𝑢→0, (3.17) as 𝑛→∞. Moreover, according to Proposition 2.7, we have ‖𝑢_𝑛−𝑢‖_∞→0, as 𝑛→∞. Observing that 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛−𝑢=−^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢_𝑛(𝑡)−𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)(𝑡)−−𝑢𝑑𝑡_𝑇0∇𝐹𝑡,𝑢_𝑛(𝑡)−𝐹(𝑡,𝑢(𝑡)),𝑢_𝑛(𝑡)(𝑡)≥|(𝑢−𝑢𝑑𝑡cos𝜋𝛼)|_𝑛−𝑢²_𝛼−^𝑇₀∇𝐹𝑡,𝑢_𝑛(𝑡)−𝐹(𝑡,𝑢(𝑡))𝑑𝑡𝑢_𝑛−𝑢_∞, (3.18) combining this with (3.17), it is easy to verify that ‖𝑢_𝑛−𝑢‖_𝛼→0, as 𝑛→∞, and hence that 𝑢_𝑛→𝑢 in 𝐸^𝛼. Thus, 𝜑 satisfies the PS condition.

Step 5. We will show that there exists another nontrivial weak solution of problem 𝑃₂.

From the mean value theorem and H(F)₁ (v), we have 𝐹(𝑡,𝑥)−𝐹(𝑡,0)=¹₀(())=∇𝐹𝑡,𝜃𝑥,𝑥𝑑𝜃₁₀(∇𝐹(𝑡,𝜃𝑥),𝜃𝑥)𝜃≤𝑑𝜃₁₀𝜇(𝑡)𝜃|𝜃𝑥|^𝛾(𝑡𝑥|𝑑𝜃=𝜇)|^𝛾1₀𝜃^𝛾−1=𝜇(𝑡𝑥|𝑑𝜃)|^𝛾𝛾, (3.19) for all |𝑥|<𝛽, a.e. 𝑡∈[0,𝑇].

It follows from the conditions H(F)₁ (iii) and all |𝑥|≥𝛽 and a.e. 𝑡∈[0,𝑇] that |𝐹(𝑡,𝑥)|≤𝑐₀|𝑡|+𝑐₀|𝑥|^𝛼0≤𝑐₀𝑥𝛽+𝑐₀|𝑥|^𝛼0≤𝑐₀𝛽^𝛼0+𝑐₀|𝑥|^𝛼0−𝛾|𝑥|^𝛾≤𝑐₀𝛽^𝛾+𝑐₀𝛽^𝛾−𝛼0|𝑥|^𝛾, (3.20) and this together with (3.19) yields that for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇], |𝐹(≤𝜇(𝑡)𝑡,𝑥)|𝛾+𝑐₀𝛽^𝛾+𝑐₀𝛽^𝛾−𝛼0|𝑥|^𝛾(𝑥)≤𝑐₁|𝑥|^𝛾, (3.21) for some positive constant 𝑐₁.

For all 𝜆>𝜆₀, ‖𝑢‖_𝛼<1, and |𝑢|_∞<1, we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹((𝑡≥|(𝑢‖𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|‖²_𝛼−𝜆𝑐₁^𝑇₀|𝑢(𝑡)|^𝛾≥|(𝑢‖𝑑𝑡cos𝜋𝛼)|‖²_𝛼−𝜆𝑐₁𝑇^(𝛼−1)/2Γ𝛼(𝛼)√2𝛼−1^𝛾∣‖𝑢‖^𝛾_𝛼. (3.22)

So, for 𝜌>0 small enough, there exists a 𝜈>0 such that 𝜑(𝑢)>𝜈,for‖𝑢‖_𝛼=𝜌, (3.23) and ‖𝑢₀‖_𝛼>𝜌. So by the mountain pass theorem (cf. Lemma 2.11), we can get 𝑢₁∈𝐸^𝛼 which satisfies 𝜑𝑢₁=𝑐>0,𝜑^′𝑢₁=0. (3.24)

Therefore, 𝑢₁ is another nontrivial critical point of 𝜑.

Remark 3.3. We can find a potential function satisfying the hypothesis of our Theorem 3.2. For great details, please see Section 4(B) in Summary.

So far, the results involved potential functions exhibiting sublinear. The next theorem concerns problems where the potential function is superlinear.

Our hypotheses on nonsmooth potential 𝐹(𝑡,𝑥) are as follows.

H(F)₂: 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ is a function such that 𝐹(𝑡,0)=0 a.e. on [0,𝑇] and satisfies the following facts:(i)for all 𝑥∈ℝ^𝑁, 𝑡↦𝐹(𝑡,𝑥) is measurable,(ii)for a.a. 𝑡∈[0,𝑇], 𝑥↦𝐹(𝑡,𝑥) is continuously differentiable,(iii)there exist 𝑐∈𝐶([0,𝑇],ℝ) and 𝛼₀>2, such that |∇𝐹(𝑡,𝑥)|,|𝐹(𝑡,𝑥)|≤𝑐(𝑡)1+(𝑡)|𝑥|^𝛼0,fora.a.𝑡∈[0,𝑇],all𝑥∈ℝ^𝑁, (3.25)(iv)there exist 𝛾>2 and 𝜇∈𝐿^∞([0,𝑇]), such that limsup_|𝑥|→0(∇𝐹(𝑡,𝑥),𝑥)|𝑥|^𝛾<𝜇(𝑡),uniformlyfora.a.𝑡∈[0,𝑇], (3.26)(v)there exist 𝜉₀∈ℝ^𝑁, 𝑡₀∈(0,𝑇), and 𝑟₀>0, such that 𝐹(𝑡,𝜉₀)>𝛿₀>0, a.e. 𝑡∈𝐵_𝑟0(𝑡₀),where 𝐵_𝑟0(𝑡₀)∶={𝑡∈[0,𝑇]∶|𝑡−𝑡₀|≤𝑟₀}⊂[0,𝑇],(vi)for a.a. 𝑡∈[0,𝑇] and all 𝑥∈ℝ^𝑁, we have 𝐹(𝑡,𝑥)≤𝜈(𝑡)with𝜈∈𝐿^𝛽([0,𝑇],ℝ),1≤𝛽<2. (3.27)

Theorem 3.4. Suppose that H(F)₂ holds. Then there exists a 𝜆₀>0 such that for each 𝜆>𝜆₀, the problem 𝑃₂ has at least two nontrivial solutions, which correspond to the two solutions of problem (𝑃).

Proof. The steps are similar to those of Theorem 3.2. In fact, we only need to modify Step 1 and Step 4 as follows: 1′ shows that 𝜑 is coercive under the condition H(F)₂ (vi); 4′ shows that there exists a second nontrivial solution under the conditions H(F)₂ (iii) and H(F)₂ (iv). Then from Steps 1′, 2, 3, and 4′ above, problem (𝑃) has at least two nontrivial solutions.
Step 1'. By H(F)₂ (vi), for all 𝑢∈𝐸^𝛼, ‖𝑢‖_𝛼>1, we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹((𝑡≥|(𝑢‖𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|‖²_𝛼−𝜆𝑐₀^𝑇₀𝜈(𝑡)𝑑𝑡→∞,as‖𝑢‖_𝛼→∞, (3.28) where 𝑐₀=max_{𝑡∈[0,𝑇]}|𝑐(𝑡)|.
Step 4'. Because of hypothesis H(F)₂ (iii), we have 𝐹(𝑡,𝑥)≤𝑐₀+𝑐₀|𝑥|^𝛼0≤𝑐₀𝑥𝛽^𝛼0+𝑐₀|𝑥|^𝛼0=𝑐₀1𝛽^𝛼0|𝑥|^𝛼0+𝑐₀|𝑥|^𝛼0=𝑐₂|𝑥|^𝛼0, (3.29) for a.e. 𝑡∈[0,𝑇] and all |𝑥|≥𝛽 with 𝑐₂>0.
Combining (3.19) and (3.29), it follows that |𝐹(≤𝜇(𝑡)𝑡,𝑥)|𝛾|𝑥|^𝛾+𝑐₂|𝑥|^𝛼0, (3.30) for a.e. 𝑡∈[0,𝑇] and all 𝑥∈ℝ^𝑁.
Thus, for all 𝜆>𝜆₀, ‖𝑢‖_𝛼<1 and |𝑢|_∞<1, we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−𝜆^𝑇₀𝐹((𝑡≥|(𝑢‖𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|‖²_𝛼−𝜆^𝑇₀𝜇(𝑡)𝛾|𝑢(𝑡)|^𝛾𝑑𝑡−𝜆𝑐₂^𝑇₀|𝑢(𝑡)|^𝛼0≥|(𝑢‖𝑑𝑡cos𝜋𝛼)|‖²_𝛼−𝜆𝑐₃𝑇^(𝛼−1)/2Γ(𝛼)√2𝛼−1^𝛾‖𝑢‖^𝛾_𝛼−𝜆𝑐₄𝑇^(𝛼−1)/2Γ(𝛼)√2𝛼−1^𝛼0‖𝑢‖^𝛼0_𝛼, (3.31) where 𝑐₃ and 𝑐₄ are positive constants.
So, for 𝜌>0 small enough, there exists a 𝜈>0 such that 𝜑(𝑢)>𝜈,for‖𝑢‖_𝛼=𝜌, (3.32) and ‖𝑢₀‖_𝛼>𝜌.
Arguing as in proof of Step 4 of Theorem 3.2, we conclude that 𝜑 satisfies the PS condition. So by the mountain pass theorem (cf. Lemma 2.11), we can get that 𝑢₁∈𝐸^𝛼 satisfies 𝜑𝑢₁=𝑐>0,𝜑^′𝑢₁=0. (3.33)
Therefore, 𝑢₁ is another nontrivial critical point of 𝜑.

Remark 3.5. We will give some examples, which satisfy the hypothesis of our Theorem 3.4. For great details, please see Section 4(C) in Summary.

4. Summary

(A) If 𝛽=0, then 𝛼=1−𝛽/2=1. Therefore, by Theorems 3.2 and 3.4, we actually obtain the existence of two weak solutions of the following eigenvalue problem: 𝑢^′′(𝑡)((𝑡[],𝑢(0)(𝑇)+𝜆∇𝐹𝑡,𝑢))=0,a.a.𝑡∈0,𝑇=𝑢=0, 𝑃₃ where 𝜆>0 is a real number, 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ is a given function, and ∇𝐹(𝑡,𝑥) is the gradient of 𝐹 at 𝑥. Although many excellent results have been worked out on the existence of solutions for second-order system 𝑃₃ (e.g., [28, 29]), it seems that no similar results were obtained in the literature for fractional system (𝑃).

(B) We give an example in the following to illustrate our viewpoint in Remark 3.1. We consider 𝐹()=𝑎(𝑡)𝜋𝑡,𝑥sin2|𝑥|^𝛾|𝑥|𝑎(𝑡),|𝑥|,0≤<1,≥1, (4.1) where 𝛾>2, 𝑎∈𝐶([0,𝑇],ℝ), 𝑎(𝑡)>0 for all 𝑡∈[0,𝑇].

Obviously, hypotheses H(F)₁ (i), (ii), and (v) are satisfied. Moreover, |𝐹(≤𝑎(𝑡)𝜋𝑡,𝑥)|2|𝑥|^𝛾(𝑡)𝜋≤𝑎2|𝑥|^𝛾−𝛼0|𝑥|^𝛼0(𝑡)𝜋≤𝑎2|𝑥|^𝛼0≤𝑎(𝑡)𝜋𝛾2|𝑥|^𝛼0|𝑥|𝑎(𝑡)(𝑡),0≤<1,≤𝑎𝜋𝛾2,|𝑥||(=𝑎(𝑡)≥1,∇𝐹𝑡,𝑥)|𝜋𝛾2𝜋cos2|𝑥|^𝛾|𝑥|^𝛾−2𝑥≤𝑎(𝑡)𝜋𝛾2|𝑥|^𝛾−1≤𝑎(𝑡)𝜋𝛾2|𝑥||𝑥|,0≤<1,0,≥1. (4.2)

Therefore, |(,|𝐹(≤(𝑡)∇𝐹𝑡,𝑥)|𝑡,𝑥)|𝜋𝛾𝑎2+𝜋𝛾𝑎(𝑡)2|𝑥|^𝛼0. (4.3) So, condition H(F)₁(iii) holds.

On the other hand, limsup_|𝑥|→0⟨∇𝐹(𝑡,𝑥),𝑥⟩|𝑥|^𝛾=limsup_|𝑥|→0𝛾(𝜋/2)𝑎(𝑡)cos(𝜋/2)|𝑥|^𝛾|𝑥|^𝛾|𝑥|^𝛾=𝜋2𝛾𝑎(𝑡), (4.4) uniformly for a.a. 𝑡∈[0,𝑇], so condition H(F)₁ (iv) holds.

(C) We can find the following potential functions satisfying the conditions stated in Theorem 3.4: 𝐹₁()=−𝑎(𝑡)𝑡,𝑥𝛾|𝑥|^𝛾,|𝑥|𝑎(𝑡)<1,𝜋𝜋cos2|𝑥|²−𝑎(𝑡)𝛾,|𝑥|𝐹≥1,₂()=𝑎(𝑡)𝜋𝑡,𝑥sin2|𝑥|^𝛾,|𝑥|𝑎(𝑡1<1,)|𝑥|+12|𝑥|²−12,|𝑥|𝐹≥1,₃()=𝑎(𝑡)𝑡,𝑥𝛾|𝑥|^𝛾,|𝑥|−(𝑡)<1,2𝑎|𝑥|(𝑡)1+𝑎2+𝛾,|𝑥|≥1, (4.5) where 𝑎∈𝐶([0,𝑇],ℝ), 𝑎(𝑡)>0 for all 𝑡∈[0,𝑇].

It is clear that 𝐹_𝑖(𝑥,0)=0 (𝑖=1,2,3) for a.e. 𝑡∈[0,𝑇], and hypotheses H(F)₂ (i) and H(F)₂ (ii) are satisfied. A direct verification shows that conditions H(F)₂ (v) and H(F)₂ (vi) are satisfied. Note that ∇𝐹₁(=(𝑡𝑥|𝑡,𝑥)−𝑎)|^𝛾−2𝑥≤𝑎(𝑡)|𝑥|^𝛾−1(𝑡)|𝑥|(𝑡)𝜋≤2𝑎,0≤<1,−𝑎sin2|𝑥|²𝑥≤𝑎(𝑡)|𝑥|≤2𝑎(𝑡)|𝑥|^𝛾,|𝑥|≥1,∇𝐹₂(=(𝑡)𝜋𝑡,𝑥)𝛾𝑎cos2|𝑥|^𝛾𝜋2|𝑥|^𝛾−2𝑥(𝑡)𝜋≤𝛾𝑎2|𝑥|^𝛾−1≤𝑎(𝑡)𝜋𝛾2|𝑥|𝑎(𝑡)𝑥,0≤<1,𝑥−|𝑥|³(𝑡)𝑥|+1≤𝑎||𝑥|²≤𝑎(𝑡)+𝑎(𝑡)|𝑥|^𝛾,|𝑥|≥1,∇𝐹₃(=𝑎(𝑡𝑥|𝑡,𝑥))|^𝛾−2𝑥≤𝑎(𝑡)|𝑥|^2(5/2)(𝑡)|𝑥|𝑎(𝑡𝑥|≤𝑎,0≤<1,)|^−5/2𝑥≤𝑎(𝑡)|𝑥|≤𝑎(𝑡)|𝑥|^𝛾,|𝑥|𝐹≥1,₁(𝑡,𝑥)≤𝑎(𝑡)|𝑥|^𝛾+𝑎(𝑡)(1+1)=2𝑎(𝑡)+2𝑎(𝑡)|𝑥|^𝛾,𝐹₂((𝑡𝜋𝑡,𝑥)≤𝑎)sin2|𝑥|^𝛾(𝑡1+𝑎)1+2|𝑥|^𝛾−12≤3𝑎(𝑡)+𝑎(𝑡)|𝑥|^𝛾2,𝐹₃((𝑡)1𝑡,𝑥)≤𝑎𝛾|𝑥|^𝛾+2𝑎(𝑡)|𝑥|(𝑡)1+𝑎2+𝛾≤5𝑎(𝑡)+𝑎(𝑡)|𝑥|^𝛾. (4.6) So, ∇𝐹₁(𝑡,𝑥),𝐹₁(𝑡,𝑥)≤2𝑎(𝑡)1+|𝑥|^𝛾,∇𝐹₂(𝑡,𝑥),𝐹₂(𝑡,𝑥)≤𝑎(𝑡)𝜋𝛾21+|𝑥|^𝛾,∇𝐹₃(𝑡,𝑥),𝐹₃(𝑡,𝑥)≤5𝑎(𝑡)1+|𝑥|^𝛾,limsup_|𝑥|→0∇𝐹₁(𝑡,𝑥),𝑥|𝑥|^𝛾=lim_|𝑥|→0−𝑎(𝑡)|𝑥|^𝛾|𝑥|^𝛾=−𝑎(𝑡),limsup_|𝑥|→0∇𝐹₂(𝑡,𝑥),𝑥|𝑥|^𝛾=lim_|𝑥|→0𝑎(𝑡)cos(𝜋/2)|𝑥|^𝛾(𝜋/2)𝛾|𝑥|^𝛾|𝑥|^𝛾=lim_|𝑥|→0𝑎(𝑡)𝜋cos2|𝑥|^𝛾𝜋2(𝑡)𝛾=𝛾𝜋𝑎2,limsup_|𝑥|→0∇𝐹₃(𝑡,𝑥),𝑥|𝑥|^𝛾=lim_|𝑥|→0𝑎(𝑡)|𝑥|^𝛾|𝑥|^𝛾=𝑎(𝑡), (4.7) which shows that assumptions H(F)₂ (iii) and H(F)₂ (iv) are fulfilled.

Acknowlegments

This work was supported by the National Natural Science Foundation of China (nos. 11126286, 11001063, and 11201095), the Fundamental Research Funds for the Central Universities (no. 2012), China Postdoctoral Science Foundation funded project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50325).