Ground State Solutions for a Class of Fractional Differential Equations with Dirichlet Boundary Value Condition

Abstract

In this paper, we apply the method of the Nehari manifold to study the fractional differential equation (𝑑/𝑑𝑡)((1/2)₀𝐷_𝑡^−𝛽(𝑢^′(𝑡))+(1/2)_𝑡𝐷_𝑇^−𝛽(𝑢^′(𝑡)))=𝑓(𝑡,𝑢(𝑡)), a.e. 𝑡∈[0,𝑇], and 𝑢(0)=𝑢(𝑇)=0, where ₀𝐷_𝑡^−𝛽,_𝑡𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 0≤𝛽<1, respectively. We prove the existence of a ground state solution of the boundary value problem.

1. Introduction

Fractional differential equations have played an important role in many fields such as engineering, science, electrical circuits, diffusion, and applied mathematics (see [1–4]). In recent years, some authors have studied the fractional differential equation by using different methods, such as fixed point theorem, coincidence degree theory, and critical point theory (see [5–22]).

By using the Mountain Pass theorem, Jiao and Zhou [17] studied the existence of solutions for the following boundary value problem: 𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12^𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇)+∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇=𝑢=0, (1) where 0<𝛽<1, ₀𝐷_𝑡^−𝛽 and _𝑡𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 𝛽, respectively, 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ, and ∇𝐹(𝑡,𝑥) is the gradient of 𝐹 with respect to 𝑥.

By using a critical-points theorem established by G. Bonanno, Bai [19] investigated the following fractional boundary value problem: 𝑑1𝑑𝑡2⁰𝐷_𝑡^𝛼−1𝐶₀𝐷^𝛼_𝑡𝑢(𝑡)−12^𝑡𝐷_𝑇^𝛼−1𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡)(𝑡)𝑓(𝑢(𝑡[],𝑢(0)(𝑇)+𝜆𝑎))=0,a.e.𝑡∈0,𝑇=𝑢=0, (2) where 𝛼∈(1/2,1], ₀𝐷_𝑡^𝛼−1 and _𝑡𝐷_𝑇^𝛼−1 are the left and right Riemann-Liouville fractional integrals of order 1−𝛼, and ^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡) and ^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡) are the left and right Caputo fractional derivatives of order 𝛼.

The authors in [18, 20–22] further studied the existence and multiplicity of solutions for the related problems by critical point theory.

We find that the method of Nehari manifold is seldom used in the above boundary value problem. Inspired by the results in [16–22], we would like to investigate the ground state solution for the following fractional boundary value problem: 𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12^𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡)((𝑡,[],𝑢(0)(𝑇)=𝑓𝑡,𝑢))a.e.𝑡∈0,𝑇=𝑢=0, (3) where 0<𝛽<1, ₀𝐷_𝑡^−𝛽 and _𝑡𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 𝛽, respectively. The technical tool is the method of Nehari manifold. (see [23, 24]).

This paper is organized as follows. In Section 2, some preliminaries on the fractional calculus are presented. In Section 3, we set up the variational framework of problem (3) and give some necessary lemmas. Finally, Section 4 presents the main result and its proof.

2. Preliminaries on the Fractional Calculus

In this section, we will introduce some notations, definitions, and preliminary facts on fractional calculus which are used throughout this paper.

Definition 1 (left and right Riemann-Liouville fractional integrals). Let 𝑓 be a function defined on [𝑎,𝑏]. The left and right Riemann-Liouville fractional integrals of order 𝛼 for function 𝑓 denoted by _𝑎𝐷_𝑡^−𝛼𝑓(𝑡) and _𝑡𝐷_𝑏^−𝛼𝑓(𝑡) function, respectively, are defined by _𝑎𝐷_𝑡^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏],𝛼>0,_𝑡𝐷_𝑏^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑏_𝑡(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏],𝛼>0, (4) provided that the right-hand side integral is pointwise defined on [𝑎,𝑏].

Definition 2 (left and right Riemann-Liouville fractional derivatives). Let 𝑓 be a function defined by [𝑎,𝑏]. The left and right Riemann-Liouville fractional derivatives of order 𝛼 for function 𝑓 denoted by _𝑎𝐷^𝛼_𝑡𝑓(𝑡) and _𝑡𝐷^𝛼_𝑏𝑓(𝑡) function, respectively, are defined by _𝑎𝐷^𝛼_𝑡𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝑎𝐷_𝑡^𝛼−𝑛𝑓(𝑡)=1Γ(𝛼)𝑑^𝑛𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,𝛼>0,_𝑡𝐷^𝛼_𝑏𝑓(𝑡)=(−1)^𝑛𝑑^𝑛𝑑𝑡^𝑛𝑡𝐷_𝑏^𝛼−𝑛𝑓(𝑡)=(−1)^𝑛Γ(𝛼)𝑑^𝑛𝑑𝑡^𝑛𝑏_𝑡(𝑠−𝑡)^{𝑛−𝛼−1}𝑓(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,𝛼>0, (5) provided that the right-hand side integral is pointwise defined on [𝑎,𝑏].

Definition 3 (left and right Caputo fractional derivatives). If 𝛼∈(𝑛−1,𝑛) and 𝑓∈𝐴𝐶^𝑛([𝑎,𝑏],ℝ), then the left and right Caputo fractional derivatives of order 𝛼 for function 𝑓 denoted by ^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡) and ^𝐶_𝑡𝐷^𝛼_𝑏𝑓(𝑡) function, respectively, are defined by ^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=_𝑎𝐷_𝑡^𝛼−𝑛𝑑^𝑛𝑑𝑡^𝑛𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,𝛼>0,_𝐶𝑡𝐷^𝛼_𝑏𝑓(𝑡)=(−1)^𝑛_𝑡𝐷_𝑏^𝛼−𝑛𝑑^𝑛𝑑𝑡^𝑛𝑓(𝑡)=(−1)^𝑛Γ(𝛼)^𝑏_𝑡(𝑠−𝑡)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)[]𝑑𝑠,𝑡∈𝑎,𝑏,𝛼>0, (6) respectively, where 𝑡∈[𝑎,𝑏].

Lemma 4 (see [18]). The left and right Riemann-Liouville fractional integral operators have the property of a semigroup; that is, ^𝑏_𝑎𝑎𝐷_𝑡^−𝛼𝑓(𝑡)𝑔(𝑡)𝑑𝑡=^𝑏_𝑎𝑡𝐷_𝑏^−𝛼𝑔(𝑡)𝑓(𝑡)𝑑𝑡,𝛼>0, (7) provided that 𝑓∈𝐿^𝑝([𝑎,𝑏],ℝ), 𝑔∈𝐿^𝑞([𝑎,𝑏],ℝ) and 𝑝≥𝑞, 𝑞≥1, 1/𝑝+1/𝑞≤1+𝛼 or 𝑝≠1, 𝑞≠1, 1/𝑝+1/𝑞=1+𝛼.

Lemma 5 (see [18]). Assume that 𝑛−1<𝛼<𝑛 and 𝑓∈𝐶^𝑛[𝑎,𝑏]. Then, _𝑎𝐷_𝑡^−𝛼𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑎)()𝑗!𝑡−𝑎^𝑗,_𝑡𝐷_𝑏^−𝛼𝐶_𝑡𝐷^𝛼_𝑏𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑗=0(−1)^𝑗𝑓^(𝑗)(𝑏)()𝑗!𝑏−𝑡^𝑗, (8) for 𝑡∈[𝑎,𝑏].

Lemma 6 (see [18]). Assume that 𝑛−1<𝛼<𝑛. Then, ^𝐶_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=_𝑎𝐷^𝛼_𝑡𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑎)Γ()𝑗−𝛼+1𝑡−𝑎^𝑗−𝛼,𝑡∈[𝑎,𝑏_𝐶𝑡],𝐷^𝛼_𝑏𝑓(𝑡)=_𝑡𝐷^𝛼_𝑏𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑏)Γ()𝑗−𝛼+1𝑏−𝑡^𝑗−𝛼,𝑡∈[𝑎,𝑏]. (9)

3. A Variational Setting

To apply critical point theory for the existence of solutions for boundary value problem (3), we shall state some basic notations and results [18], which will be used in the proof of our main results.

Throughout this paper, we denote 𝛼=1−𝛽/2 and assume that the following conditions are satisfied.(H1)𝑓∈𝐶¹(ℝ×ℝ).(H2)𝑓(𝑡,0)=0=(𝜕𝑓/𝜕𝑠)(𝑡,0) for every 𝑡∈ℝ.(H3)There are constants 𝑎,𝑏>0 and 2<𝑝 such that 𝜕𝑓(|𝑠|𝜕𝑠𝑡,𝑠)≤𝑎+𝑏^𝑝−2, (10) for every 𝑡∈ℝ and 𝑠∈ℝ.(H4)There are constants 𝜇>2,𝑀>0 such that 0<𝜇𝐹(𝑡,𝑠)≤𝑠𝑓(𝑡,𝑠) (11) for all 𝑡∈ℝ and |𝑠|≥𝑀. Here, 𝐹(𝑡,𝑠)=^𝑠₀𝑓(𝑡,𝑥)𝑑𝑥. (12)(H5)The map 𝑡→𝑡⁻¹𝑠𝑓(𝑥,𝑡𝑠) is increasing on (0,+∞), for every 𝑥∈ℝ and 𝑠∈ℝ.

Now we construct appropriate function spaces. Denote by 𝐶^∞₀([0,𝑇],ℝ) the set of all functions 𝑢∈𝐶^∞₀([0,𝑇],ℝ) with 𝑢(0)=𝑢(𝑇)=0. The fractional derivative space 𝐸₀^𝛼,𝑝 is defined by the closure of 𝐶^∞₀([0,𝑇],ℝ) with respect to the norm ‖𝑢‖_𝛼,𝑝=^𝑇₀|𝑢(𝑡)|^𝑝𝑑𝑡+^𝑇₀^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡)^𝑝𝑑𝑡^1/𝑝, (13) where ^𝐶₀𝐷^𝛼_𝑡 is the 𝛼-order left Caputo fractional derivative.

Remark 7. If 𝑝=2, we define 𝐸^𝛼=𝐸₀^𝛼,2, with respect to the norm ‖𝑢‖=^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡)²𝑑𝑡^1/2. (14) The set 𝐸^𝛼 is a reflexive and separable Hilbert space.

Remark 8. For any 𝑢∈𝐸^𝛼, noting the fact 𝑢(0)=0, we have ₀𝐷^𝛼_𝑡𝑢(𝑡)=^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡),𝑡∈[0,𝑇].

Lemma 9 (see [17]). 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. (15) 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. (16)

According to (15), we can consider 𝐸^𝛼 with respect to the equivalent norm ‖𝑢‖_𝛼,𝑝=^𝐶₀𝐷^𝛼_𝑡𝑢_𝐿𝑝,‖𝑢‖=^𝐶₀𝐷^𝛼_𝑡𝑢_𝐿2. (17)

Lemma 10 (see [17]). Let 0<𝛼≤1 and 1<𝑝<∞. Assume that 𝛼>1/𝑝 and the sequence {𝑢_𝑘} converges weakly to 𝑢 in 𝐸₀^𝛼,𝑝; that is, 𝑢_𝑘⇀𝑢. Then 𝑢_𝑘→𝑢 in 𝐶([0,𝑇],ℝ); that is, ‖𝑢−𝑢_𝑘‖_∞→0 as 𝑘→∞.

Similar to the proof of [17, Proposition 4.1], we have the following property.

Lemma 11. If 1/2<𝛼≤1, for any 𝑢∈𝐸^𝛼, one has |cos(𝜋𝛼)|‖𝑢‖²≤−^𝑇₀^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡)≤1𝑑𝑡|(‖𝑢‖cos𝜋𝛼)|². (18)

To obtain a weak solution of boundary value problem (3), we assume that 𝑢 is a sufficiently smooth solution of (3). Multiplying (3) by an arbitrary 𝑣∈𝐶^∞₀(0,𝑇), we have −^𝑇₀𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12^𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡)(𝑡)=,𝑣𝑑𝑡_𝑇0𝑓(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡. (19)

Observe that −12^𝑇₀𝑑𝑑𝑡⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+_𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡)(𝑡)=1,𝑣𝑑𝑡2^𝑇₀⁰𝐷_𝑡^−𝛽𝑢^′(𝑡),𝑣^′(𝑡)+_𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡),𝑣^′(𝑡=1)𝑑𝑡2^𝑇₀⁰𝐷_𝑡^−𝛽/2𝑢^′(𝑡),_𝑡𝐷_𝑇^−𝛽/2𝑣^′(𝑡)+_𝑡𝐷_𝑇^−𝛽/2𝑢^′(𝑡),₀𝐷_𝑡^−𝛽/2𝑣^′(𝑡)𝑑𝑡. (20) As 𝑢(0)=𝑢(𝑇)=𝑣(0)=𝑣(𝑇)=0, we have ₀𝐷_𝑡^−𝛽/2𝑢^′(𝑡)=₀𝐷_𝑡^1−𝛽/2𝑢(𝑡),_𝑡𝐷_𝑇^−𝛽/2𝑢^′(𝑡)=−_𝑡𝐷_𝑇^1−𝛽/2𝑢(𝑡),₀𝐷_𝑡^−𝛽/2𝑣^′(𝑡)=₀𝐷_𝑡^1−𝛽/2𝑣(𝑡),_𝑡𝐷_𝑇^−𝛽/2𝑣^′(𝑡)=−_𝑡𝐷_𝑇^1−𝛽/2𝑣(𝑡). (21) Then (19) is equivalent to ^𝑇₀−12⁰𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑣(𝑡)+_𝑡𝐷^𝛼_𝑇𝑢(𝑡),₀𝐷^𝛼_𝑡𝑣(𝑡)=𝑑𝑡_𝑇0𝑓(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡. (22) Since (22) is well defined for 𝑢,𝑣∈𝐸^𝛼, the weak solution of (3) can be defined as follows.

Definition 12. A weak solution of (3) is a function 𝑢∈𝐸^𝛼 such that ^𝑇₀−12⁰𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑣(𝑡)+_𝑡𝐷^𝛼_𝑇𝑢(𝑡),₀𝐷^𝛼_𝑡𝑣(𝑡)=𝑑𝑡_𝑇0𝑓(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡 (23) for every 𝑣∈𝐸^𝛼.

We consider the functional 𝐼∶𝐸^𝛼→ℝ, defined by 𝐼(𝑢)=^𝑇₀−12⁰𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑢(𝑡)−𝐹(𝑡,𝑢(𝑡))𝑑𝑡. (24) From Theorem 4.1 of [17], we can get that if 1/2<𝛼≤1, then the functional 𝐼 is continuously differentiable on 𝐸^𝛼. Since 𝐼 is continuously differentiable on 𝐸^𝛼, then 𝐼^′(𝑢),𝑣=−^𝑇₀12⁰𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑣(𝑡)+_𝑡𝐷^𝛼_𝑇𝑢(𝑡),₀𝐷^𝛼_𝑡𝑣(𝑡)−𝑑𝑡_𝑇0𝑓(𝑡,𝑢(𝑡)),𝑣(𝑡)𝑑𝑡 (25) for 𝑢,𝑣∈𝐸^𝛼. Hence, a critical point of 𝐼 is a weak solution of problem (3).

4. Main Result

In order to study the solvability of boundary value problem (3), we use the so-called Nehari method. Define 𝐼(𝑢)1=−2^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡,∀𝑢∈𝐸^𝛼, (26) where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)𝑑𝑠.

There is one-to-one correspondence between the critical points of 𝐼 and weak solutions of boundary value problem (3). Now, we define 𝒩=𝑢∈𝐸^𝛼⧵{0}∣𝐼^′(𝑢),𝑢=0. (27) Then we know any nonzero critical point of 𝐼 must be on 𝒩. Define 𝐺(𝑢)=𝐼^′(𝑢),𝑢=−^𝑇₀₀𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑢(𝑡)−𝑑𝑡_𝑇0𝑓(𝑡,𝑢(𝑡)),𝑢(𝑡)𝑑𝑡. (28)

Lemma 13. Assume the hypotheses (𝐻₁)–(𝐻₅) hold. If 𝑢∈𝒩 is a critical point of 𝐼|_𝒩, then 𝐼^′(𝑢)=0.

Proof. For 𝑢∈𝒩, together with (H₅), 𝐺^′(𝑢),𝑢=−^𝑇₀2₀𝐷^𝛼_𝑡𝑢(𝑡),_𝑡𝐷^𝛼_𝑇𝑢(𝑡)+𝜕𝑓((𝑡𝜕𝑢𝑡,𝑢))⋅𝑢²(𝑡)((𝑡𝑢(𝑡)=+𝑓𝑡,𝑢))𝑑𝑡_𝑇0((𝑡𝑢(𝑡)−𝜕2𝑓𝑡,𝑢))𝑓((𝑡𝜕𝑢𝑡,𝑢))⋅𝑢²(𝑡)((𝑡𝑢(𝑡)=−𝑓𝑡,𝑢))𝑑𝑡_𝑇0𝑓((𝑡𝑢(𝑡)−𝜕𝑡,𝑢))𝑓((𝑡𝜕𝑢𝑡,𝑢))⋅𝑢²(𝑡)𝑑𝑡<0. (29) If 𝑢∈𝒩 is a critical point of 𝐼|_𝒩, there exists a Lagrange multiplier 𝜆∈ℝ, such that 𝐼^′(𝑢)=𝜆𝐺^′(𝑢). Then we have 𝐼^′(𝑢),𝑢=𝜆𝐺^′(𝑢),𝑢=0. (30) By (29), ⟨𝐺^′(𝑢),𝑢⟩≠0, and we have 𝜆=0. So we can get that 𝐼^′(𝑢)=0. The proof is complete.

Lemma 14. Assume the hypotheses (𝐻₁)–(𝐻₅) hold. For any 𝑢∈𝐸^𝛼⧵{0}, there is a unique 𝑦=𝑦(𝑢) such that 𝑦(𝑢)𝑢∈𝒩 and one has 𝐼(𝑦𝑢)=max_𝑦≥0𝐼(𝑦𝑢)>0.

Proof. First, we claim that there exist constants 𝛿>0, 𝜌>0 such that 𝐼(𝑢)>0 for all 𝑢∈𝐵_𝜌(0)⧵{0} and 𝐼(𝑢)≥𝛿 for all 𝑢∈𝜕𝐵_𝜌(0). That is, 0 is a strict local minimizer of 𝐼. In fact, by (H₃) we can get that ∀𝜀>0,∃𝐶_𝜀|𝐹(≤𝜀>0,𝑡,𝑢)|2|𝑢|²+𝐶_𝜀|𝑢|^𝑝. (31) Then together with Lemmas 9 and 11, we have 𝐼(𝑢)1=−2^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−^𝑇₀𝐹((𝑡1𝑡,𝑢))𝑑𝑡≥−2^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡−𝜀)𝑑𝑡2^𝑇₀|𝑢|²𝑑𝑡−𝐶_𝜀^𝑇₀|𝑢|^𝑝≥1𝑑𝑡2|cos(𝜋𝛼)|‖𝑢‖²−𝜀2^𝑇₀|𝑢|²𝑑𝑡−𝐶_𝜀^𝑇₀|𝑢|^𝑝≥1𝑑𝑡2|(−𝜀cos𝜋𝛼)|2𝑇^2𝛼Γ²()‖𝑢‖𝛼+1²−𝐶_𝜀𝑇^{𝑝+𝛼−1/2}Γ(𝛼)[(𝛼−1)2+1]^1/2𝑝‖𝑢‖^𝑝. (32) Choose 𝜀 such that (𝜀/2)(𝑇^2𝛼/Γ²(𝛼+1))=(1/4)|cos(𝜋𝛼)|; then 𝐼(𝑢)≥14|cos(𝜋𝛼)|‖𝑢‖²−𝐶_𝜀𝑇^{𝑝+𝛼−1/2}Γ(𝛼)[(𝛼−1)2+1]^1/2𝑝‖𝑢‖^𝑝=‖𝑢‖²14|cos(𝜋𝛼)|−𝐶_𝜀𝑇^{𝑝+𝛼−1/2}Γ(𝛼)[(𝛼−1)2+1]^1/2𝑝‖𝑢‖^𝑝−2. (33) Choose 𝜌>0, such that 𝐶_𝜀(𝑇^{𝑝+𝛼−1/2}/Γ(𝛼)[(𝛼−1)2+1]^1/2)^𝑝𝜌^𝑝−2=(1/8)|cos(𝜋𝛼)|. Then we have 𝐼(𝑢)≥(1/8)|cos(𝜋𝛼)|‖𝑢‖². Let 𝛿=(1/8)|cos(𝜋𝛼)|‖𝑢‖²; then we get that there exist constants 𝛿>0, 𝜌>0 such that 𝐼(𝑢)>0 for all 𝑢∈𝐵_𝜌(0)⧵{0} and 𝐼(𝑢)≥𝛿 for all 𝑢∈𝜕𝐵_𝜌(0).
Next, we claim that 𝐼(𝑦𝑢)→−∞, as 𝑦→∞. In fact, by (H₄), there exists a constant 𝐴>0 such that 𝐹(𝑡,𝑢)≥𝐴|𝑢|^𝜇 for |𝑢|≥𝑀. On the other hand, we can easily get that there exists a constant 𝐵 such that 𝐹(𝑡,𝑢)≥𝐵 for |𝑢|≤𝑀. Then together with Lemma 11, we have 𝐼≤𝑦𝑦𝑢²2|(‖𝑢‖cos𝜋𝛼)|²−𝐴𝑦^𝜇𝑇₀|𝑢|^𝜇𝑑𝑡−𝐵. (34) Since 𝜇>2, we can get that 𝐼(𝑦𝑢)→−∞, as 𝑦→∞.
Let 𝑔(𝑦)∶=𝐼(𝑦𝑢) for 𝑦>0. From what we have proved, there has at least one 𝑦_𝑢=𝑦(𝑢)>0 such that 𝑔𝑦_𝑢=max_𝑦≥0𝑔𝑦=max_𝑦≥0𝐼𝑦𝑢=𝐼𝑦_𝑢𝑢. (35) We prove next 𝑔(𝑦) has a unique critical point for 𝑦>0. Consider a critical point 𝑔^′𝑦=𝐼^′𝑦𝑢,𝑢=−^𝑇₀𝑦₀𝐷^𝛼_𝑡𝑢,_𝑡𝐷^𝛼_𝑇𝑢𝑑𝑡−^𝑇₀𝑓𝑡,𝑦𝑢𝑢𝑑𝑡=0. (36) Then, together with (H₅), we have 𝑔^′′𝑦=−^𝑇₀₀𝐷^𝛼_𝑡𝑢,_𝑡𝐷^𝛼_𝑇𝑢𝑑𝑡−^𝑇₀𝜕𝑓𝑡,𝑦𝑢𝜕𝑢𝑦𝑢²𝑑𝑡=^𝑇₀𝑓𝑡,𝑦𝑢𝑢𝑦𝑑𝑡−^𝑇₀𝜕𝑓𝑡,𝑦𝑢𝜕𝑢𝑦𝑢²=1𝑑𝑡𝑦^2𝑇₀𝑓𝑡,𝑦𝑢𝑦𝑢𝑑𝑡−^𝑇₀𝜕𝑓𝑡,𝑦𝑢𝜕𝑢𝑦𝑢²𝑑𝑡<0. (37) So we know that if 𝑦 is a critical point of 𝑔, then it must be a strict local maximum. This implies the uniqueness.
Finally, from 𝑔^′𝑦=𝐼^′=1𝑦𝑢,𝑢𝑦𝐼^′𝑦𝑢,𝑦𝑢, (38) we see 𝑦 is a critical point if 𝑦𝑢∈𝒩. Define 𝑚=inf_𝒩𝐼. Then we can get that 𝑚≥inf_{𝜕𝐵𝜌(0)}𝐼≥𝛿>0. The proof is complete.

Lemma 15. Assume the hypotheses (𝐻₁)–(𝐻₅) hold and 𝑚=inf_𝒩𝐼. Then there exists 𝑢∈𝒩 such that 𝐼(𝑢)=𝑚.

Proof. We claim that both 𝐼 and 𝐺 are weakly lower semicontinuous. In fact, according to Lemma 10, if 𝑢_𝑘⇀𝑢 in 𝐸^𝛼, then 𝑢_𝑘→𝑢 in 𝐶([0,𝑇],ℝ). Therefore, 𝐹(𝑡,𝑢_𝑘(𝑡))→𝐹(𝑡,𝑢(𝑡)) a.e. 𝑡∈[0,𝑇]. By the Lebesgue dominated convergence theorem, we have ∫^𝑇₀𝐹(𝑡,𝑢_𝑘(𝑡))𝑑𝑡→∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡, which means that the functional 𝑢→∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡 is weakly continuous on 𝐸^𝛼. Similarly 𝑢→∫^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑢(𝑡)𝑑𝑡 is weakly continuous on 𝐸^𝛼. Since 𝐸^𝛼 is Hilbet space, together with (17) and Lemma 11, we can easily get that −∫^𝑇₀(^𝐶₀𝐷^𝛼_𝑡𝑢(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢(𝑡))𝑑𝑡 is weakly lower semicontinuous on 𝐸^𝛼. Then we get that both 𝐼 and 𝐺 are weakly lower semicontinuous.
Since 𝜇𝐹(𝑡,𝑢)−𝑢𝑓(𝑡,𝑢) is continuous for 𝑡∈[0,𝑇] and |𝑥|≤𝑀, there exists 𝐵>0, such that 𝐹()≤1𝑡,𝑢𝜇𝑢𝑓(𝑡,𝑢)+𝐵,𝑡∈[0,𝑇],|𝑥|≤𝑀. (39) Together with (H₄), we get 𝐹()≤1𝑡,𝑢𝜇𝑢𝑓(𝑡,𝑢)+𝐵,𝑡∈[0,𝑇],𝑥∈ℝ. (40) Let {𝑢_𝑘}∈𝒩 be a minimizing sequence; that is, 𝐼(𝑢_𝑘)→𝑚, 𝐼^′(𝑢_𝑘)→0 as 𝑘→∞. Then, 𝑚+𝑜(1)=𝐼𝑢_𝑘1=−2^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢_𝑘(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢_𝑘(𝑡)−𝑑𝑡_𝑇0𝐹𝑡,𝑢_𝑘(𝑡)1𝑑𝑡≥−2^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢_𝑘(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢_𝑘(𝑡)−1𝑑𝑡𝜇^𝑇₀𝑢_𝑘𝑓𝑡,𝑢_𝑘=1𝑑𝑡−𝐵𝑇𝜇−12^𝑇₀^𝐶0𝐷^𝛼_𝑡𝑢_𝑘(𝑡),^𝐶_𝑡𝐷^𝛼_𝑇𝑢_𝑘(𝑡)+1𝑑𝑡𝜇𝐼^′𝑢_𝑘,𝑢_𝑘≥1−𝐵𝑇2−1𝜇|cos(𝜋𝛼)|𝑢_𝑘²−1𝜇𝐼^′𝑢_𝑘𝑢_𝑘−𝐵𝑇. (41) By 𝜇>2 and 𝐼^′(𝑢_𝑘)→0, we get that 𝑢_𝑘 is bounded in 𝐸. Since 𝐸^𝛼 is a reflexive space, going to a subsequence if necessary, we may assume that 𝑢_𝑘⇀𝑢 in 𝐸^𝛼. Then from Lemma 10, 𝑢_𝑘→𝑢 in 𝐶([0,𝑇],ℝ). Since 𝐺 is weakly lower semicontinuous and {𝑢_𝑘}∈𝒩, we first have 𝐺(𝑢)≤lim_𝑘→∞𝐺𝑢_𝑘=0. (42) Then we have 𝑢≠0. In fact, if 𝑢=0, then 𝑢_𝑘→0 in 𝐶([0,𝑇],ℝ). By 𝐺(𝑢_𝑘)=0, we get ‖𝑢_𝑘‖→0. This is a contradiction with {𝑢_𝑘}∈𝒩.
Then from Lemma 14, there exists a unique 𝑦>0 such that 𝑦𝑢∈𝒩. Together with 𝐼 which is weakly lower semicontinuous, we have 𝑚≤𝐼𝑦𝑢≤lim_𝑘→∞𝐼𝑦𝑢_𝑘≤lim_𝑘→∞𝐼𝑦𝑢_𝑘≤lim_𝑘→∞𝐼𝑢_𝑘=𝑚. (43) Then we get that 𝑚 is achieved at 𝑦𝑢∈𝒩. The proof is complete.

Theorem 16. Assuming the hypotheses (𝐻₁)–(𝐻₅) hold, boundary value problem (3) has a weak solution such that 𝐼(𝑢)=𝑚; that is, boundary value problem (3) has a ground state solution.

Proof. By the Lemmas 14 and 15, we can get that there exists 𝑢∈𝒩 such that 𝐼(𝑢)=𝑚=inf_𝒩𝐼. Then the 𝑢 is a critical point of 𝐼|_𝒩. From Lemma 13 we have 𝐼^′(𝑢)=0. So boundary value problem (3) has a weak solution such that 𝐼(𝑢)=𝑚. The proof is complete.

5. Example

In this section, we give an example to illustrate our results.

Example 1. Consider the following BVP: −𝑑1𝑑𝑡2⁰𝐷_𝑡^−1/2+12^𝑡𝐷_𝑇^−1/2𝑢^′(𝑡)=𝑢³[],𝑢(0)(𝑇),𝑡∈0,𝑇=0,𝑢=0. (44) It is easy to verify all the conditions in Theorem 16, so BVP (44) has a ground state solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by the Fundamental Research Funds for the Central Universities (2013XK03).