Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory

Abstract

We study the existence and multiplicity of solutions for the following fractional boundary value problem: (𝑑/𝑑𝑡)((1/2)₀𝐷_𝑡^−𝛽(𝑢′(𝑡))+(1/2)_𝑡𝐷_𝑇^−𝛽(𝑢′(𝑡)))+∇𝐹(𝑡,𝑢(𝑡))=0,a.e.𝑡∈[0,𝑇],𝑢(0)=𝑢(𝑇)=0, where 𝐹(𝑡,⋅) are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.

1. Introduction and Main Results

Consider the fractional boundary value problem (BVP for short) of the following form: 𝑑1𝑑𝑡2⁰𝐷_𝑡^−𝛽𝑢^′(𝑡)+12^𝑡𝐷_𝑇^−𝛽𝑢^′(𝑡)((𝑡[],𝑢(0)(𝑇)+∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇=𝑢=0, (1.1) where ₀𝐷_𝑡^−𝛽 and _𝑡𝐷_𝑇^−𝛽 are the left and right Riemann-Liouville fractional integrals of order 0≤𝛽<1, respectively, 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ satisfies the following assumptions.(A)𝐹(𝑡,𝑥) is measurable in 𝑡 for every 𝑥∈ℝ^𝑁 and continuously differentiable in 𝑥 for a.e. 𝑡∈[0,𝑇], and there exist 𝑎∈𝐶(ℝ⁺,ℝ⁺), 𝑏∈𝐿¹(0,𝑇;ℝ⁺), such that |𝐹(𝑡,𝑥)|≤𝑎(|𝑥|)𝑏(𝑡),|∇𝐹(𝑡,𝑥)|≤𝑎(|𝑥|)𝑏(𝑡), (1.2) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇]. In particular, if 𝛽=0, BVP (1.1) reduces to the standard second-order boundary value problem of the following form: 𝑢^′′(𝑡)((𝑡[],𝑢(0)(𝑇)+∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇=𝑢=0. (1.3)

Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not mere mathematical curiosities but rather has interesting applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Such differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Samko et al. [4], and the papers [5–20] and the references therein.

Recently, there are many papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or singular and nonsingular boundary) value problems of fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems [12–14], Leray-Schauder theory [15, 16], lower and upper solution method, monotone iterative method [17–19], Adomian decomposition method [20], etc.), see [12–20] and the references therein.

Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis. The existence and multiplicity of solutions for Hamilton systems, Schrödinger equations, and Dirac equations have been studied extensively via critical point theory, see [21–34].

In [32], Jiao and Zhou obtained the existence of solutions for BVP (1.1) by Mountain Pass theorem under the Ambrosetti-Rabinowitz condition (denoted by A.R. condition). Under the usual A.R. condition, it is easy to show that the energy functional associated with the system has the Mountain Pass geometry and satisfies the (PS) condition. However, the A.R. condition is so strong that many potential functions cannot satisfy it, then the problem becomes more delicate and complicated.

In this paper, in order to establish the existence and multiplicity of solutions for BVP (1.1) under distinct hypotheses on potential function by critical point theory, we introduce some functional space 𝐸^𝛼, where 𝛼∈(1/2,1], and divide the problem into the following three cases.

1.1. The Superquadratic Case

For the superquadratic case, we make the following assumptions. (A1)lim_|𝑥|→0𝐹(𝑡,𝑥)/|𝑥|²=0, liminf_|𝑥|→∞𝐹(𝑡,𝑥)/|𝑥|²≥𝐿>𝜋²/|cos(𝜋𝛼)|Γ²(2−𝛼)𝑇^2𝛼(3−2𝛼) uniformly for some 𝐿>0 and a.e. 𝑡∈[0,𝑇].(A2)limsup_{|𝑥|→+∞}𝐹(𝑡,𝑥)/|𝑥|^𝑟≤𝑀<+∞ uniformly for some 𝑀>0 and a.e. 𝑡∈[0,𝑇].(A3)liminf_{|𝑥|→+∞}((∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥))/|𝑥|^𝜇≥𝑄>0 uniformly for some 𝑄>0 and a.e. 𝑡∈[0,𝑇],

where 𝑟>2 and 𝜇>𝑟−2. We state our first existence result as follows.

Theorem 1.1. Assume that (A1)–(A3) hold and that 𝐹(𝑡,𝑥) satisfies the condition (A). Then BVP (1.1) has at least one solution on 𝐸^𝛼.

1.2. The Asymptotically Quadratic Case

For the asymptotically quadratic case, we assume the following.(A2′)limsup_{|𝑥|→+∞}𝐹(𝑡,𝑥)/|𝑥|²≤𝑀<+∞ uniformly for some 𝑀>0 and a.e. 𝑡∈[0,𝑇].(A4)There exists 𝜏(𝑡)∈𝐿¹(0,𝑇;ℝ⁺) such that (∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)≥𝜏(𝑡) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇].(A5)lim_{|𝑥|→+∞}[(∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)]=+∞ for a.e. 𝑡∈[0,𝑇].

Our second and third main results read as follows.

Theorem 1.2. Assume that 𝐹(𝑡,𝑥) satisfies (A), (A1), (A2'), (A4), and (A5). Then BVP (1.1) has at least one solution on 𝐸^𝛼.

Theorem 1.3. Assume that 𝐹(𝑡,𝑥) satisfies (A), (A1), (A2'), and the following conditions: (A4′) there exists 𝜏(𝑡)∈𝐿¹(0,𝑇;ℝ⁺) such that (∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)≤𝜏(𝑡) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇];(A5′)lim_{|𝑥|→+∞}[(∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)]=−∞ for a.e. 𝑡∈[0,𝑇].Then BVP (1.1) has at least one solution on 𝐸^𝛼.

1.3. The Subquadratic Case

For the subquadratic case, we give the following multiplicity result.

Theorem 1.4. Assume that 𝐹(𝑡,𝑥) satisfies the following assumption: (A6)𝐹(𝑡,𝑥)∶=𝑎(𝑡)|𝑥|^𝛾, where 𝑎(𝑡)∈𝐿^∞(0,𝑇;ℝ⁺) and 1<𝛾<2 is a constant.

Then BVP (1.1) has infinitely many solutions on 𝐸^𝛼.

2. Preliminaries

In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space 𝐸^𝛼 are also listed for the convenience of readers.

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

Definition 2.2 (see [1]). Let 𝑓(𝑡) be a function defined on [𝑎,𝑏] and 𝑞>0. The left and right Riemann-Liouville fractional derivatives of order 𝑞 for function 𝑓(𝑡) denoted by _𝑎𝐷^𝑞_𝑡𝑓(𝑡) and _𝑡𝐷^𝑞_𝑏𝑓(𝑡), respectively, are defined by _𝑎𝐷^𝑞_𝑡𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝑎𝐷_𝑡^𝑞−𝑛𝑓(𝑡)=1Γ𝑑𝑛−𝑞𝑑𝑡^𝑛𝑛𝑡_𝑎(𝑡−𝑠)^{𝑛−𝑞−1}𝑓(𝑠)𝑑𝑠_𝑡,𝐷^𝑞_𝑏𝑓(𝑡)=(−1)^𝑛𝑑^𝑛𝑑𝑡^𝑛𝑡𝐷_𝑏^𝑞−𝑛𝑓(𝑡)=1Γ()𝑛−𝑞−1^𝑛𝑑^𝑛𝑑𝑡^𝑛𝑏_𝑡(𝑠−𝑡)^{𝑛−𝑞−1}𝑓(𝑠)𝑑𝑠, (2.2) where 𝑡∈[𝑎,𝑏], 𝑛−1≤𝑞<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 belonging to the space of absolutely continuous functions, which we denote by 𝐴𝐶([𝑎,𝑏],ℝ^𝑁). 𝐴𝐶^𝑘([𝑎,𝑏],ℝ^𝑁)(𝑘=1,…) is the space of functions 𝑓 such that 𝑓∈𝐶^𝑘−1([𝑎,𝑏],ℝ^𝑁) and 𝑓^(𝑘−1)∈𝐴𝐶([𝑎,𝑏],ℝ^𝑁). In particular, 𝐴𝐶([𝑎,𝑏],ℝ^𝑁)=𝐴𝐶¹([𝑎,𝑏],ℝ^𝑁).

Definition 2.3 (see [1]). Let 𝑞≥0 and 𝑛∈ℕ. If 𝑞∈[𝑛−1,𝑛) and 𝑓(𝑡)∈𝐴𝐶^𝑛([𝑎,𝑏],ℝ^𝑁), then the left and right Caputo fractional derivative of order 𝑞 for function 𝑓(𝑡) denoted by ^𝑐_𝑎𝐷^𝑞_𝑡𝑓(𝑡) and ^𝑐_𝑡𝐷^𝑞_𝑏𝑓(𝑡), respectively, exist almost everywhere on [𝑎,𝑏]. ^𝑐_𝑎𝐷^𝑞_𝑡𝑓(𝑡) and ^𝑐_𝑡𝐷^𝑞_𝑏𝑓(𝑡) are represented by ^𝑐_𝑎𝐷^𝑞_𝑡𝑓(𝑡)=_𝑎𝐷_𝑡^𝑞−𝑛𝑓^(𝑛)(𝑡)=1Γ𝑛−𝑞^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝑞−1}𝑓^(𝑛)(𝑠)𝑑𝑠_𝑐𝑡,𝐷^𝑞_𝑏𝑓(𝑡)=(−1)^𝑛_𝑡𝐷_𝑏^𝑞−𝑛𝑓^(𝑛)(𝑡)=(−1)^𝑛Γ𝑛−𝑞^𝑏_𝑡(𝑠−𝑡)^{𝑛−𝑞−1}𝑓^(𝑛)(𝑠)𝑑𝑠, (2.3)(2.4) respectively, where 𝑡∈[𝑎,𝑏].

Property 2.4 (see [1]). The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, that is, _𝑎𝐷^−𝑞1_𝑡𝑎𝐷^−𝑞2_𝑡𝑓(𝑡)=_𝑎𝐷^{−𝑞1−𝑞2}_𝑡𝑓(𝑡),_𝑡𝐷^−𝑞1_𝑏𝑡𝐷^−𝑞2_𝑏𝑓(𝑡)=_𝑡𝐷^{−𝑞1−𝑞2}_𝑏𝑓(𝑡),∀𝑞₁,𝑞₂>0. (2.5)

Definition 2.5 (see [32]). 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, (2.6) 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 2.6 (see [32]). Let 0<𝛼≤1 and 1<𝑝<∞. The fractional derivative space 𝐸₀^𝛼,𝑝 is a reflexive and separable space.

Proposition 2.7 (see [32]). Let 0<𝛼≤1 and 1<𝑝<∞. For all 𝑢∈𝐸₀^𝛼,𝑝, one has ‖𝑢‖_𝐿𝑝≤𝑇^𝛼Γ(𝛼+1)^𝑐0𝐷^𝛼_𝑡𝑢_𝐿𝑝. (2.7) 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. (2.8)

According to (2.8), we can consider 𝐸₀^𝛼,𝑝 with respect to the norm ‖𝑢‖_𝛼,𝑝=^𝑐₀𝐷^𝛼_𝑡𝑢_𝐿𝑝=^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡)^𝑝𝑑𝑡^1/𝑝. (2.9)

Proposition 2.8 (see [32]). Define 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^𝑁), that is, ‖𝑢−𝑢_𝑘‖_∞→0, as 𝑘→∞.

Making use of Property 2.4 and Definition 2.3, for any 𝑢∈𝐴𝐶([0,𝑇],ℝ^𝑁), BVP (1.1) 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)+∇𝐹𝑡,𝑢))=0,a.e.𝑡∈0,𝑇=𝑢=0, (2.10) where 𝛼=1−𝛽/2∈(1/2,1].

In the following, we will treat BVP (2.10) in the Hilbert space 𝐸^𝛼=𝐸₀^𝛼,2 with the corresponding norm ‖𝑢‖_𝛼=‖𝑢‖_𝛼,2. The variational structure of BVP (2.10) on the space 𝐸^𝛼 has been established.

Lemma 2.9 (see [32]). Let 𝐿∶[0,𝑇]×ℝ^𝑁×ℝ^𝑁×ℝ^𝑁→ℝ be defined by 𝐿1𝑡,𝑥,𝑦,𝑧=−2𝑦,𝑧−𝐹(𝑡,𝑥), (2.11) where 𝐹∶[0,𝑇]×ℝ^𝑁→ℝ satisfies the assumption (A).

If 1/2<𝛼≤1, then the functional defined by 𝜑(𝑢)=^𝑇₀𝐿𝑡,𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡 (2.12) is continuously differentiable on 𝐸^𝛼, and ∀𝑢,𝑣∈𝐸^𝛼, we have 𝜑^′(𝑢),𝑣=^𝑇₀𝐷_𝑥𝐿𝑡,𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)(𝑡)+,𝑣𝑑𝑡_𝑇0𝐷_𝑦𝐿𝑡,𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑣(𝑡)+𝑑𝑡_𝑇0𝐷_𝑧𝐿𝑡,𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑣(𝑡)𝑑𝑡. (2.13)

Definition 2.10 (see [32]). A function 𝑢∈𝐴𝐶([0,𝑇],ℝ^𝑁) is called a solution of BVP (2.10) if (i)𝐷^𝛼(𝑢(𝑡)) is derivative for almost every 𝑡∈[0,𝑇], (ii)𝑢 satisfies (2.10), 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)).

Lemma 2.11 (see [32]). Let 1/2<𝛼≤1 and 𝜑 be defined by (2.12). If assumption (A) is satisfied and 𝑢∈𝐸^𝛼 is a solution of corresponding Euler equation 𝜑^′(𝑢)=0, then 𝑢 is a solution of BVP (2.10) which corresponding to the solution of BVP (1.1).

By Lemma 2.11, it means that the solutions for BVP (1.1) correspond to the critical points of the functional 𝜑. We need the following estimate and known results for the sequel.

Proposition 2.12 (see [32]). 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. (2.14)

Lemma 2.13 (see [23]). Let 𝑋 be a real Banach space, Φ∶𝑋→ℝ is differentiable. One says that Φ satisfies the (PS) condition if any sequence {𝑢_𝑘} in 𝑋 such that {Φ(𝑢_𝑘)} is bounded and Φ^′(𝑢_𝑘)→0 as 𝑘→∞ contains a convergent subsequence.

Lemma 2.14 (Mountain Pass theorem [24]). Let 𝑋 be a real Banach space and Φ∶𝑋→ℝ is differentiable and satisfies the (PS) condition. Suppose that (i)Φ(0)=0,(ii)there exist 𝜌>0 and 𝜎>0 such that Φ(𝑧)≥𝜎 for all 𝑧∈𝑋 with ‖𝑧‖=𝜌, (iii)there exists 𝑧₁ in 𝑋 with ‖𝑧₁‖≥𝜌 such that Φ(𝑧₁)<𝜎.Then Φ possesses a critical value 𝑐≥𝜎. Moreover, 𝑐 can be characterized as 𝑐=inf_𝑔∈Ωmax_{𝑧∈𝑔([0,1])}Φ(𝑧), (2.15) where Ω={𝑔∈𝐶([0,1],𝑋)∶𝑔(0)=0,𝑔(1)=𝑧₁}.

Lemma 2.15 (Clark theorem  [24]). Let 𝑋 be a real Banach space, Φ∈𝐶¹(𝑋,ℝ) with Φ even, bounded below, and satisfying the (PS) condition. Suppose Φ(0)=0, there is a set 𝐾⊂𝑋 such that 𝐾 is homeomorphic to 𝑆^𝑚−1, 𝑚∈ℕ, by an odd map, and sup_𝐾Φ<0. Then Φ possesses at least 𝑚 distinct pairs of critical points.

3. Proof of the Theorems

For 𝑢∈𝐸^𝛼, where 𝐸^𝛼∶=𝑢∈𝐿²0,𝑇;ℝ^𝑁∶^𝑐₀𝐷^𝛼_𝑡𝑢∈𝐿²0,𝑇;ℝ^𝑁 (3.1) is a reflexive Banach space with the norm defined by ‖𝑢‖_𝛼=^𝑐₀𝐷^𝛼_𝑡𝑢_𝐿2,‖𝑢‖_∞∶=max_{[]𝑡∈0,𝑇}|𝑢(𝑡)|. (3.2)

It follows from Lemma 2.9 that the functional 𝜑 on 𝐸^𝛼 given by 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)−𝐹(𝑡,𝑢(𝑡))𝑑𝑡 (3.3) is continuously differentiable on 𝐸^𝛼. Moreover, 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_𝑇0(∇𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡. (3.4)

Recall that a sequence {𝑢_𝑛}∈𝐸^𝛼 is said to be a (C) sequence of 𝜑 if 𝜑(𝑢_𝑛) is bounded and (1+‖𝑢_𝑛‖_𝛼)‖𝜑(𝑢_𝑛)‖_𝛼→0 as 𝑛→∞. The functional 𝜑 satisfies condition (C) if every (C) sequence of 𝜑 has a convergent subsequence. This condition is due to Cerami [21].

3.1. Proof of Theorem 1.1

We will first establish the following lemma and then give the proof of Theorem 1.1.

Lemma 3.1. Assume (A), (A2), and (A3) hold, then the functional 𝜑 satisfies condition (C).

Proof of Lemma 3.1. Let {𝑢_𝑛}⊂𝐸^𝛼 be a (C) sequence of 𝜑, that is, 𝜑(𝑢_𝑛) is bounded and (1+‖𝑢_𝑛‖_𝛼)‖𝜑′(𝑢_𝑛)‖_𝛼→0 as 𝑛→∞. Then there exists 𝑀₀ such that 𝜑𝑢_𝑛≤𝑀₀,1+𝑢_𝑛𝛼𝜑^′𝑢_𝑛𝛼≤𝑀₀, (3.5) for all 𝑛∈ℕ.
By (A2), there exist positive constants 𝐵₁ and 𝑀₁ such that 𝐹(𝑡,𝑥)≤𝐵₁|𝑥|^𝑟, (3.6) for all |𝑥|≥𝑀₁ and a.e. 𝑡∈[0,𝑇].
It follows from (A) that |𝐹(𝑡,𝑥)|≤max_{𝑠∈0,𝑀1}𝑎(𝑠)𝑏(𝑡), (3.7) for all |𝑥|≤𝑀₁ and a.e. 𝑡∈[0,𝑇]. Therefore, we obtain 𝐹(𝑡,𝑥)≤𝐵₁|𝑥|^𝑟+max_{𝑠∈0,𝑀1}𝑎(𝑠)𝑏(𝑡), (3.8) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇].
Combining (2.14) and (3.8), we get |cos(𝜋𝛼)|2𝑢_𝑛²_𝛼≤𝜑𝑢_𝑛+^𝑇₀𝐹𝑡,𝑢_𝑛(𝑡)𝑑𝑡≤𝑀₀+max_{𝑠∈0,𝑀1}𝑎(𝑠)^𝑇₀𝑏(𝑡)𝑑𝑡+𝐵₁^𝑇₀𝑢_𝑛(𝑡)^𝑟𝑑𝑡. (3.9)
On the other hand, by (A3), there exist 𝜂>0 and 𝑀₂>0 such that (∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)≥𝜂|𝑥|^𝜇, (3.10) for a.e. 𝑡∈[0,𝑇] and |𝑥|≥𝑀₂.
By (A), we have |(∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)|≤2+𝑀₂max_{𝑠∈0,𝑀2}𝑎(𝑠)𝑏(𝑡), (3.11) for all |𝑥|≤𝑀₂ and a.e. 𝑡∈[0,𝑇].
Therefore, we obtain (∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)≥𝜂|𝑥|^𝜇−𝜂𝑀^𝜇₂−2+𝑀₂max_{𝑠∈0,𝑀2}𝑎(𝑠)𝑏(𝑡), (3.12) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇].
It follows from (3.5) and (3.12) that 3𝑀₀≥2𝜑𝑢_𝑛−𝜑^′𝑢_𝑛,𝑢_𝑛=2^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢_𝑛(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)−𝐹𝑡,𝑢_𝑛(𝑡)−𝑑𝑡_𝑇0−^𝑐₀𝐷^𝛼_𝑡𝑢_𝑛(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)−∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)=𝑑𝑡_𝑇0∇𝐹𝑡,𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)−2𝐹𝑡,𝑢_𝑛(𝑡)𝑑𝑡≥𝜂^𝑇₀𝑢_𝑛(𝑡)^𝜇𝑑𝑡−2+𝑀₂max_{𝑠∈[0,𝑀2]}𝑎(𝑠)^𝑇₀𝑏(𝑡)𝑑𝑡−𝜂𝑀^𝜇₂𝑇, (3.13) thus, ∫^𝑇₀|𝑢_𝑛(𝑡)|^𝜇𝑑𝑡 is bounded.
If 𝜇>𝑟, then ^𝑇₀𝑢_𝑛(𝑡)^𝑟𝑑𝑡≤𝑇^{(𝜇−𝑟)/𝜇𝑇}₀𝑢_𝑛(𝑡)^𝜇𝑑𝑡^𝑟/𝜇, (3.14) which, combining (3.9), implies that ‖𝑢_𝑛‖_𝛼 is bounded.
If 𝜇≤𝑟, then ^𝑇₀𝑢_𝑛(𝑡)^𝑟𝑑𝑡≤𝑢_𝑛∞^{𝑟−𝜇𝑇}₀𝑢_𝑛(𝑡)^𝜇𝑑𝑡≤𝐶₁^𝑟−𝜇𝑢_𝑛𝛼^{𝑟−𝜇𝑇}₀𝑢_𝑛(𝑡)^𝜇𝑑𝑡, (3.15) where 𝐶₁𝑇∶=^𝛼−1/2Γ(𝛼)(2𝛼−1)^1/2, (3.16) by (2.8).
Since 𝜇>𝑟−2, it follows from (3.9) that ‖𝑢_𝑛‖_𝛼 is bounded too. Thus ‖𝑢_𝑛‖_𝛼 is bounded in 𝐸^𝛼.
By Proposition 2.8, the sequence {𝑢_𝑛} has a subsequence, also denoted by {𝑢_𝑛}, such that 𝑢_𝑛⇀𝑢weaklyin𝐸^𝛼,𝑢_𝑛⟶𝑢stronglyin𝐶[0,𝑇],ℝ^𝑁. (3.17)
Then we obtain 𝑢_𝑛→𝑢 in 𝐸^𝛼 by use of the same argument of Theorem  5.2 in [32]. The proof of Lemma 3.1 is completed.

Proof of Theorem 1.1. By (A1), there exist 𝜖₁∈(0,|cos(𝜋𝛼)|) and 𝛿>0 such that 𝐹(𝑡,𝑥)≤|cos(𝜋𝛼)|−𝜖₁Γ²(𝛼+1)2𝑇^2𝛼|𝑥|², (3.18) for a.e. 𝑡∈[0,𝑇] and 𝑥∈ℝ^𝑁 with |𝑥|≤𝛿.
Let Γ(𝛼2())𝜌=)(𝛼−1+1^1/2𝑇^𝛼−1/2𝜖𝛿,𝜎=¹𝜌²2>0. (3.19) Then it follows from (2.8) that ‖𝑢‖_∞≤𝑇^𝛼−1/2Γ(𝛼)(2(𝛼−1)+1)^1/2‖𝑢‖_𝛼=𝛿, (3.20) for all 𝑢∈𝐸^𝛼 with ‖𝑢‖_𝛼=𝜌.
Therefore, we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)((𝑡≥|(−𝐹𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−|cos(𝜋𝛼)|−𝜖₁Γ²(𝛼+1)2𝑇^2𝛼𝑇₀|𝑢(𝑡)|²≥|(𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−|cos(𝜋𝛼)|−𝜖₁2‖𝑢‖²_𝛼=𝜖₁2‖𝑢‖²_𝛼=𝜎, (3.21) for all 𝑢∈𝐸^𝛼 with ‖𝑢‖_𝛼=𝜌. This implies that (ii) in Lemma 2.14 is satisfied.
It is obvious from the definition of 𝜑 and (A1) that 𝜑(0)=0, and therefore, it suffices to show that 𝜑 satisfies (iii) in Lemma 2.14.
By (A1), there exist 𝜖₂>0 and 𝑀₃>0 such that 𝐹()>𝜋𝑡,𝑥²|cos(𝜋𝛼)|Γ²(2−𝛼)𝑇^2𝛼(3−2𝛼)+𝜖₂|𝑥|², (3.22) for all |𝑥|≥𝑀₃ and a.e. 𝑡∈[0,𝑇].
It follows from (A) that |𝐹(𝑡,𝑥)|≤max_{𝑠∈0,𝑀3}𝑎(𝑠)𝑏(𝑡), (3.23) for all |𝑥|≤𝑀₃ and a.e. 𝑡∈[0,𝑇].
Therefore, we obtain 𝐹()≥𝜋𝑡,𝑥²|cos(𝜋𝛼)|Γ²(2−𝛼)𝑇^2𝛼(3−2𝛼)+𝜖₂|𝑥|²−𝑀²₃−max_{𝑠∈0,𝑀3}𝑎(𝑠)𝑏(𝑡), (3.24) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇].
Choosing 𝑢₀=((𝑇/𝜋)sin(𝜋𝑡/𝑇),0,…,0)∈𝐸^𝛼, then 𝑢₀²_𝐿2=𝑇³2𝜋²,𝑢₀²_𝛼≤𝑇^3−2𝛼Γ²(2−𝛼)(3−2𝛼). (3.25)
For 𝜍>0 and noting that (3.24) and (3.25), we have 𝜑𝜍𝑢₀=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝜍𝑢₀(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝜍𝑢₀(𝑡)−𝐹𝑡,𝜍𝑢₀(𝑡)≤𝜍𝑑𝑡₂2|(𝑢cos𝜋𝛼)|^02𝛼−𝜍²𝜋²|cos(𝜋𝛼)|𝑇^2𝛼Γ²(2−𝛼)(3−2𝛼)+𝜍²𝜖₂^𝑇₀𝑢₀(𝑡)²𝑑𝑡+𝐶₂≤𝜍²2|(⋅𝑇cos𝜋𝛼)|Γ²(^3−2𝛼)−𝜍2−𝛼)(3−2𝛼²𝜋²|cos(𝜋𝛼)|𝑇^2𝛼Γ²()⋅𝑇2−𝛼)(3−2𝛼2𝜋³²−𝜍²𝜖₂𝑇³2𝜋²+𝐶₂⟶−∞, (3.26) as 𝜍→∞, where 𝐶₂ is a positive constant. Then there exists a sufficiently large 𝜍₀ such that 𝜑(𝜍₀𝑢₀)≤0. Hence (iii) holds.
Finally, noting that 𝜑(0)=0 while for critical point 𝑢, 𝜑(𝑢)≥𝜎>0. Hence 𝑢 is a nontrivial solution of BVP (1.1), and this completes the proof.

3.2. Proof of Theorem 1.2

The following lemmata are needed in the proof of Theorem 1.2.

Lemma 3.2. Assume (A5), then for any 𝜀>0, there exists a subset 𝐸_𝜀⊂[0,𝑇] with meas([0,𝑇]⧵𝐸_𝜀)<𝜀 such that lim_|𝑥|→∞[(∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)]=+∞, (3.27) uniformly for 𝑡∈𝐸_𝜀.

Proof of Lemma 3.2. The proof is similar to that of Lemma  2 in [29] and is omitted.

Lemma 3.3. Assume (A), (A2’), (A4), and (A5), then the functional 𝜑 satisfies condition (C).

Proof of Lemma 3.3. Suppose that {𝑢_𝑛}⊂𝐸^𝛼 is a (C) sequence of 𝜑, that is, 𝜑(𝑢_𝑛) is bounded and (1+‖𝑢_𝑛‖_𝛼)‖𝜑^′(𝑢_𝑛)‖_𝛼→0 as 𝑛→∞. Then we have liminf_𝑛→∞𝜑^′𝑢_𝑛,𝑢_𝑛−2𝜑𝑢_𝑛>−∞, (3.28) which implies that limsup_𝑛→∞^𝑇₀∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛𝑑𝑡<+∞. (3.29)
We only need to show that {𝑢_𝑛} is bounded in 𝐸^𝛼. If {𝑢_𝑛} is unbounded, we may assume, without loss of generality, that ‖𝑢_𝑛‖_𝛼→∞ as 𝑛→∞. Put 𝑧_𝑛=𝑢_𝑛/‖𝑢_𝑛‖_𝛼, we then have ‖𝑧_𝑛‖_𝛼=1. Going to a sequence if necessary, we assume that 𝑧_𝑛⇀𝑧 weakly in 𝐸^𝛼, 𝑧_𝑛→𝑧 strongly in 𝐶([0,𝑇],ℝ^𝑁) and 𝐿²(0,𝑇;ℝ^𝑁).
By (A2), it follows that there exist constants 𝐵₂>0 and 𝑀₄>0 such that 𝐹(𝑡,𝑥)≤𝐵₂|𝑥|², (3.30) for all |𝑥|≥𝑀₄ and a.e. 𝑡∈[0,𝑇].
By assumption (A), it follows that |𝐹(𝑡,𝑥)|≤max_{𝑠∈0,𝑀4}𝑎(𝑠)𝑏(𝑡), (3.31) for all |𝑥|≤𝑀₄ and a.e. 𝑡∈[0,𝑇]. Therefore, we obtain 𝐹(𝑡,𝑥)≤𝐵₂|𝑥|²+max_{𝑠∈0,𝑀4}𝑎(𝑠)𝑏(𝑡) (3.32) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇]. Therefore, we have 𝜑(𝑢)=^𝑇₀−12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)((𝑡≥|(−𝐹𝑡,𝑢))𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−𝐵₂^𝑇₀|𝑢|²𝑑𝑡−max_{𝑠∈0,𝑀4}𝑎(𝑠)^𝑇₀𝑏(𝑡)𝑑𝑡, (3.33) from which, it follows that 𝜑𝑢_𝑛𝑢_𝑛²_𝛼≥|cos(𝜋𝛼)|2−𝐵₂𝑧_𝑛²_𝐿2−1𝑢_𝑛²_𝛼max_{𝑠∈0,𝑀4}𝑎(𝑠)^𝑇₀𝑏(𝑡)𝑑𝑡. (3.34)
Passing to the limit in the last inequality, we get |cos(𝜋𝛼)|2−𝐵₂‖𝑧‖²_𝐿2≤0, (3.35) which yields 𝑧≠0. Therefore, there exists a subset 𝐸⊂[0,𝑇] with meas(𝐸)>0 such that 𝑧(𝑡)≠0 on 𝐸.
By virtue of Lemma 3.2, for 𝜀=(1/2) meas(𝐸)>0, we can choose a subset 𝐸_𝜀⊂[0,𝑇] with meas([0,𝑇]⧵𝐸_𝜀)<𝜀 such that lim_|𝑥|→∞[(∇𝐹(𝑡,𝑥),𝑥)−2𝐹(𝑡,𝑥)]=+∞, (3.36) uniformly for 𝑡∈𝐸_𝜀.
We assert that meas(𝐸⋂𝐸_𝜀)>0. If not, meas(𝐸⋂𝐸_𝜀)=0.
Since 𝐸=(𝐸⋂𝐸_𝜀)⋃(𝐸⧵𝐸_𝜀), it follows that 0<meas(𝐸)=meas𝐸∩𝐸_𝜀+meas𝐸⧵𝐸_𝜀≤meas[0,𝑇]⧵𝐸_𝜀1<𝜀=2meas(𝐸), (3.37) which leads to a contradiction and establishes the assertion.
By (A4), we obtain thye following: ^𝑇₀∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛=𝑑𝑡_{𝐸⋂𝐸𝜀}∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛𝑑𝑡+_{⋂𝐸[0,𝑇]⧵(𝐸𝜀)}∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛≥𝑑𝑡_{𝐸⋂𝐸𝜀}∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛𝑑𝑡−^𝑇₀|𝜏(𝑡)|𝑑𝑡. (3.38)
By (3.36), (3.38), and Fatou's lemma, it follows that lim_𝑛→∞^𝑇₀∇𝐹𝑡,𝑢_𝑛,𝑢_𝑛−2𝐹𝑡,𝑢_𝑛𝑑𝑡=+∞, (3.39) which contradicts (3.29). This contradiction shows that ‖𝑢_𝑛‖_𝛼 is bounded in 𝐸^𝛼, and this completes the proof.
By virtue of Lemmas 3.2 and 3.3, the rest of the proof is similar to Theorem 1.1. Theorem 1.3 can be proved similarly.

3.3. Proof of Theorem 1.4

The proof of Theorem 1.4 is divided into a sequence of lemma.

Lemma 3.4. The functional 𝜑 is bounded below on 𝐸^𝛼.

Proof of Lemma 3.4. By (2.8) and (2.14), for every 𝑢∈𝐸^𝛼, we have 𝜑(𝑢)=−^𝑇₀12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡=−^𝑇₀12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−^𝑇₀𝑎(𝑡)|𝑢(𝑡)|^𝛾≥|(𝑑𝑡cos𝜋𝛼)|2‖𝑢‖²_𝛼−𝑎₀‖𝑢‖^𝛾_∞𝑇≥|cos(𝜋𝛼)|2‖𝑢‖²_𝛼−𝑎₀𝑇𝐶^𝛾₁‖𝑢‖^𝛾_𝛼, (3.40) where 𝑎₀=esssup{𝑎(𝑡)∶𝑡∈[0,𝑇]}. The proof of Lemma 3.4 is complete.

Lemma 3.5. The functional 𝜑 satisfies the (PS) condition.

Proof of Lemma 3.5. Let {𝑢_𝑛} be a Palais-Smale sequence in 𝐸^𝛼, that is, 𝜑𝑢_𝑛isboundedand𝜑^′𝑢_𝑛⟶0as𝑛⟶+∞. (3.41)
Suppose that {𝑢_𝑛} is unbounded in 𝐸^𝛼, that is, ‖𝑢_𝑛‖_𝛼→+∞ as 𝑛→+∞. Since 𝜑^′𝑢_𝑛,𝑢_𝑛−𝛾𝜑𝑢_𝑛=𝛾−1+2^𝑇₀^𝑐0𝐷^𝛼_𝑡𝑢_𝑛(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢_𝑛(𝑡)𝑑𝑡. (3.42)
However, from (3.42), we have −𝛾𝜑𝑢_𝑛≥𝛾1−2|cos(𝜋𝛼)|𝑢_𝑛²_𝛼−𝜑^′𝑢_𝑛𝑢_𝑛𝛼, (3.43) thus ‖𝑢_𝑛‖_𝛼 is a bounded sequence in 𝐸^𝛼. Since 𝐸^𝛼 is a reflexive space, going, if necessary, to a subsequence, we can assume that 𝑢_𝑛⇀𝑢 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_𝑛𝛼𝑢_𝑛−𝑢_𝛼−𝜑^′(𝑢),𝑢_𝑛−𝑢⟶0, (3.44) as 𝑛→∞. Moreover, according to (2.8) and Proposition 2.8, we have that {𝑢_𝑛} is bounded in 𝐶([0,𝑇],ℝ^𝑁) and ‖𝑢_𝑛−𝑢‖_∞→0 as 𝑛→∞.
Noting that 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛−𝑢=−^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢_𝑛(𝑡)−𝑢(𝑡),^𝑐₀𝐷^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_∞. (3.45) Combining (3.44) and (3.45), it is easy to verify that ‖𝑢_𝑛−𝑢‖_𝛼→0 as 𝑛→∞, and hence that 𝑢_𝑛→𝑢 in 𝐸^𝛼. Thus, {𝑢_𝑛} admits a convergent subsequence. The proof of Lemma 3.5 is complete.

Lemma 3.6. For any 𝑚∈ℕ, there exists a set 𝐾⊂𝐸^𝛼 which is homeomorphic to 𝑆^𝑚−1 by an odd map, and sup_𝑘𝜑<0.

Proof of Lemma 3.6. For every 𝑚∈ℕ, define 𝑢_𝑖(𝑡)=sin𝑖𝜋𝑡𝑇𝐸,0,…,0,𝑖=1,2,…,𝑚,_𝑚=span𝑢₁,…,𝑢_𝑚,𝐾_𝑚,𝛽=𝑢∈𝐸_𝑚∶‖𝑢‖_𝛼=𝛽, (3.46) where 𝛽 is a positive number to be chosen later.
For any 𝑢∈𝐸_𝑚, there exist 𝜆_𝑖∈ℝ, 𝑖=1,2,…,𝑚, such that 𝑢=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡),‖𝑢‖²_𝛼=^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡)²𝑑𝑡=^𝑇₀^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢(𝑡)𝑑𝑡=^𝑇₀𝜆₁^𝑐₀𝐷^𝛼_𝑡𝑢₁(𝑡)+⋯+𝜆_𝑚^𝑐₀𝐷^𝛼_𝑡𝑢_𝑚(𝑡),𝜆₁^𝑐₀𝐷^𝛼_𝑡𝑢₁(𝑡)+⋯+𝜆_𝑚^𝑐₀𝐷^𝛼_𝑡𝑢_𝑚(𝑡)=𝑑𝑡_{𝑚𝑚𝑖=1𝑗=1}𝑎_𝑖𝑗𝜆_𝑖𝜆_𝑗=𝐹𝜆₁,…,𝜆_𝑚, (3.47) where 𝑎_𝑖𝑗=∫^𝑇₀(^𝑐₀𝐷^𝛼_𝑡𝑢_𝑖(𝑡),^𝑐₀𝐷^𝛼_𝑡𝑢_𝑗(𝑡))𝑑𝑡 and 𝐹(𝜆₁,…,𝜆_𝑚) is a real quadratic form.
Since 𝐹𝜆₁,…,𝜆_𝑚=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)²_𝛼≥0,∀𝜆₁,…,𝜆_𝑚^𝑇∈ℝ^𝑚,𝐹𝜆₁,…,𝜆_𝑚=0⟺^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖(𝑡)≡0⟺𝜆₁=𝜆₂=⋯=𝜆_𝑚=0. (3.48) So, 𝐹(𝜆₁,…,𝜆_𝑚) is a real positive definite quadratic form. Then there exist an invertible matrix 𝐶∈ℝ^𝑚×𝑚 and 𝜇_𝑖∈ℝ, 𝑖=1,2,…,𝑚, such that 𝜆₁,𝜆₂,…,𝜆_𝑚^𝑇=𝐶𝜇₁,𝜇₂,…,𝜇_𝑚^𝑇,𝐹𝜆₁,…,𝜆_𝑚=^𝑚_𝑖=1𝜇²_𝑖. (3.49)
It is easy to prove that the odd mapping Ψ∶𝐾_𝑚,𝛽→𝑆^𝑚−1 defined by Ψ(𝑢)=𝛽⁻¹𝜇₁,…,𝜇_𝑚 (3.50) is a homeomorphism between 𝐾_𝑚,𝛽 and 𝑆^𝑚−1.
Since 𝐸_𝑚⊂𝐸^𝛼 is a finite dimensional space, there exists 𝜀(𝑚)>0 such that meas𝑡∈[0,𝑇]∶𝑎(𝑡)|𝑢(𝑡)|^𝛾≥𝜀‖𝑢‖^𝛾_𝛼≥𝜀,∀𝑢∈𝐸_𝑚⧵{0}. (3.51)
Otherwise, for any positive integer 𝑛, there exists 𝑢_𝑛∈𝐸_𝑚⧵{0} such that meas𝑡∈[0,𝑇]∶𝑎(𝑡)𝑢_𝑛(𝑡)^𝛾≥1𝑛𝑢_𝑛^𝛾_𝛼<1𝑛. (3.52)
Set 𝑣_𝑛(𝑡)∶=𝑢_𝑛(𝑡)/‖𝑢_𝑛‖_𝛼∈𝐸_𝑚⧵{0}, then ‖𝑣_𝑛‖_𝛼=1 for all 𝑛∈ℕ and meas𝑡∈[0,𝑇]∶𝑎(𝑡)𝑣_𝑛(𝑡)^𝛾≥1𝑛<1𝑛. (3.53)
Since dim𝐸_𝑚<∞, it follows from the compactness of the unit sphere of 𝐸_𝑚 that there exists a subsequence, denoted also by {𝑣_𝑛}, such that {𝑣_𝑛} converges to some 𝑣₀ in 𝐸_𝑚. It is obvious that ‖𝑣₀‖_𝛼=1.
By the equivalence of the norms on the finite dimensional space, we have 𝑣_𝑛→𝑣₀ in 𝐿²(0,𝑇;ℝ^𝑁), that is, ^𝑇₀𝑣_𝑛−𝑣₀²𝑑𝑡⟶0as𝑛⟶∞. (3.54)
By (3.54) and Hölder inequality, we have ^𝑇₀𝑎(𝑡)𝑣_𝑛−𝑣₀^𝛾𝑑𝑡≤^𝑇₀𝑎(𝑡)^2/(2−𝛾)𝑑𝑡^{(2−𝛾)/2𝑇}₀𝑣_𝑛−𝑣₀²𝑑𝑡^𝛾/2=‖𝑎‖_(2−𝛾)/2^𝑇₀𝑣_𝑛−𝑣₀²𝑑𝑡^𝛾/2⟶0,as𝑛⟶∞. (3.55)
Thus, there exist 𝜉₁,𝜉₂>0 such that meas𝑡∈[0,𝑇]∶𝑎(𝑡)𝑣₀(𝑡)^𝛾≥𝜉₁≥𝜉₂. (3.56)
In fact, if not, we have meas𝑡∈[0,𝑇]∶𝑎(𝑡)𝑣₀(𝑡)^𝛾≥1𝑛=0, (3.57) for all positive integer 𝑛.
It implies that 0≤^𝑇₀𝑎(𝑡)𝑣₀^𝛾+2𝑇𝑑𝑡<𝑛𝑣₀²_∞≤𝐶²₁𝑇𝑛𝑣₀²_𝛼⟶0, (3.58) as 𝑛→∞. Hence 𝑣₀=0 which contradicts that ‖𝑣₀‖_𝛼=1. Therefore, (3.56) holds.
Now let Ω₀=𝑡∈[0,𝑇]∶𝑎(𝑡)𝑣₀(𝑡)^𝛾≥𝜉₁,Ω_𝑛=𝑡∈[0,𝑇]∶𝑎(𝑡)𝑣_𝑛(𝑡)^𝛾<1𝑛, (3.59) and Ω^𝑐_𝑛=[0,𝑇]⧵Ω_𝑛={𝑡∈[0,𝑇]∶𝑎(𝑡)|𝑣_𝑛(𝑡)|^𝛾≥1/𝑛}.
By (3.53) and (3.56), we have measΩ_𝑛∩Ω₀=measΩ₀⧵Ω^𝑐_𝑛∩Ω₀≥measΩ₀−measΩ^𝑐_𝑛∩Ω₀≥𝜉₂−1𝑛, (3.60) for all positive integer 𝑛. Let 𝑛 be large enough such that 𝜉₂−1𝑛≥12𝜉₂,12^𝛾−1𝜉₁−1𝑛≥12^𝛾𝜉₁, (3.61) then we have ^𝑇₀𝑎(𝑡)𝑣_𝑛−𝑣₀^𝛾𝑑𝑡≥_Ω𝑛∩Ω0𝑎(𝑡)𝑣_𝑛−𝑣₀^𝛾≥1𝑑𝑡2^𝛾−1_Ω𝑛∩Ω0𝑎(𝑡)𝑣₀^𝛾𝑑𝑡−_Ω𝑛∩Ω0𝑎(𝑡)𝑣_𝑛^𝛾≥1𝑑𝑡2^𝛾−1𝜉₁−1𝑛measΩ_𝑛∩Ω₀≥𝜉₁2^𝛾⋅𝜉₂2=𝜉₁𝜉₂2^𝛾+1>0, (3.62) for all large 𝑛, which is a contradiction to (3.55). Therefore, (3.51) holds.
For any 𝑢∈𝐾_𝑚,𝛽, we have 𝜑(𝑢)=−^𝑇₀12^𝑐0𝐷^𝛼_𝑡𝑢(𝑡),^𝑐_𝑡𝐷^𝛼_𝑇𝑢(𝑡)𝑑𝑡−^𝑇₀𝐹((𝑡≤1𝑡,𝑢))𝑑𝑡2|(‖𝑢‖cos𝜋𝛼)|^2𝛼−^𝑇₀𝑎(𝑡)|𝑢(𝑡)|^𝛾≤1𝑑𝑡2|(‖𝑢‖cos𝜋𝛼)|^2𝛼−𝜀‖𝑢‖^𝛾_𝛼measΩ_𝑢≤12|(‖𝑢‖cos𝜋𝛼)|^2𝛼−𝜀²‖𝑢‖^𝛾_𝛼, (3.63) by (3.51), where Ω_𝑢∶={𝑡∈[0,𝑇]∶𝑎(𝑡)|𝑢(𝑡)|^𝛾≥𝜀‖𝑢‖^𝛾_𝛼}.
Choosing 𝛽=(|cos(𝜋𝛼)|𝜀²)^1/(2−𝛾), we conclude sup_{𝐾𝑚,𝛽}𝜑<−𝜀²𝛽^𝛾/2<0 which completes the proof.

Now from the assertion of Lemma 2.15, we know that 𝜑 has at least 𝑚 distinct pairs of critical points for every 𝑚∈ℕ, therefore, BVP (1.1) possesses infinitely many solutions on 𝐸^𝛼. The proof of Theorem 1.4 is completed.

4. Examples

In this section, we give some examples to illustrate our results.

Example 4.1. In BVP (1.1), let 𝐹(𝑡,𝑥)=ln1+2|𝑥|²|𝑥|². (4.1)
These show that all conditions of Theorem 1.1 are satisfied, where 𝑟=2.5,𝜇=2. (4.2)
By Theorem 1.1, BVP (1.1) has at least one solution 𝑢∈𝐸^𝛼.

Example 4.2. In BVP (1.1), let 𝑇=2𝜋 and 𝐹(𝑡,𝑥)=𝜅𝑓(𝑥)(2+sin𝑡)arctan|𝑥|², where 𝜅>0 and 𝑓(𝑥) will be specified below.
Let 𝑓(𝑥)=|𝑥|²+ln(1+|𝑥|²). Noting that 0≤ln(1+|𝑥|²)≤|𝑥|², we see that (A) and (A2′) hold. It is also easy to see that (A1) holds for ()𝜅>2𝜋^1−2𝛼|cos(𝜋𝛼)|Γ²(2−𝛼)(3−2𝛼). (4.3) Furthermore, we have (𝑥)(𝑥)=2|𝑥|∇𝑓,𝑥−2𝑓²1+|𝑥|²−2ln1+|𝑥|²⟶−∞, (4.4) as |𝑥|→+∞. Therefore, we have (())()2|𝑥|∇𝐹𝑡,𝑥,𝑥−2𝐹𝑡,𝑥=𝜅²1+|𝑥|⁴𝑓(𝑥)(2+sin𝑡)+𝜅∇𝑓(𝑥),𝑥−2𝑓(𝑥)(2+sin𝑡)arctan|𝑥|²⟶−∞, (4.5) uniformly for all 𝑡∈[0,2𝜋] as |𝑥|→+∞. Thus (A4′) and (A5′) hold. By virtue of Theorem 1.3, we conclude that BVP (1.1) has at least one solution on 𝐸^𝛼.
If 𝑓(𝑥)=|𝑥|²−ln(1+|𝑥|²), then exactly the same conclusions as above hold true by Theorem 1.2.

Example 4.3. In BVP (1.1), let 𝐹(𝑡,𝑥)=𝑎(𝑡)|𝑥|^3/2 where 𝑎(𝑡)=𝑇𝑇,𝑡=02𝑡,0<𝑡≤2(),𝑇−2𝑡−𝑇2<𝑡<𝑇𝑇,𝑡=𝑇. (4.6)
By Theorem 1.4, BVP (1.1) has infinite solutions on 𝐸^𝛼.

Acknowledgment

This paper is partially supported by the NNSF (no. 11171351) of China.