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

Authors: Jing Chen; X. H. Tang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the existence and multiplicity of solutions for the following fractional boundary value problem:(d/dt)((1/2)0Dt-β(u′(t))+(1/2)tDT-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.


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## Body

## 1. Introduction and Main Results

Consider the fractional boundary value problem (BVP for short) of the following form:(1.1)ddt(120Dt-β(u′(t))+12tDT-β(u′(t)))+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,
where 0Dt-β and tDT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively,  F:[0,T]×ℝN→ℝ satisfies the following assumptions.(A)
F(t,x) is measurable in t for every x∈ℝN 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(1.2)|F(t,x)|≤a(|x|)b(t),|∇F(t,x)|≤a(|x|)b(t),
for all x∈ℝN 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: (1.3)u′′(t)+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0.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 Eα, 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)T2α(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],wherer>2 and μ>r-2. We state our first existence result as follows.Theorem 1.1.
Assume that (A1)–(A3) hold and thatF(t,x) satisfies the condition (A). Then BVP (1.1) has at least one solution on Eα.


### 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 existsτ(t)∈L1(0,T;ℝ+) such that (∇F(t,x),x)-2F(t,x)≥τ(t) for all x∈ℝN    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 thatF(t,x) satisfies (A), (A1), (A2'), (A4), and (A5). Then BVP (1.1) has at least one solution on Eα.Theorem 1.3.
Assume thatF(t,x) satisfies (A), (A1), (A2'), and the following conditions:  (A4′)
there existsτ(t)∈L1(0,T;ℝ+) such that (∇F(t,x),x)-2F(t,x)≤τ(t) for all x∈ℝN 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 Eα.


### 1.3. The Subquadratic Case

For the subquadratic case, we give the following multiplicity result.Theorem 1.4.
Assume thatF(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 Eα.


## 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)T2α(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],wherer>2 and μ>r-2. We state our first existence result as follows.Theorem 1.1.
Assume that (A1)–(A3) hold and thatF(t,x) satisfies the condition (A). Then BVP (1.1) has at least one solution on Eα.


## 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 existsτ(t)∈L1(0,T;ℝ+) such that (∇F(t,x),x)-2F(t,x)≥τ(t) for all x∈ℝN    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 thatF(t,x) satisfies (A), (A1), (A2'), (A4), and (A5). Then BVP (1.1) has at least one solution on Eα.Theorem 1.3.
Assume thatF(t,x) satisfies (A), (A1), (A2'), and the following conditions:  (A4′)
there existsτ(t)∈L1(0,T;ℝ+) such that (∇F(t,x),x)-2F(t,x)≤τ(t) for all x∈ℝN 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 Eα.


## 1.3. The Subquadratic Case

For the subquadratic case, we give the following multiplicity result.Theorem 1.4.
Assume thatF(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 Eα.


## 2. Preliminaries

In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of spaceEα are also listed for the convenience of readers.Definition 2.1 (see [1]).
Letf(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 aDt-qf(t) and tDb-qf(t), respectively, are defined by
(2.1)aDt-qf(t)=1Γ(q)∫at(t-s)q-1f(s)ds,tDb-qf(t)=1Γ(q)∫tb(t-s)q-1f(s)ds,
provided the right-hand sides are pointwise defined on [a,b], where Γ is the gamma function.Definition 2.2 (see [1]).
Letf(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 aDtqf(t) and tDbqf(t), respectively, are defined by
(2.2)aDtqf(t)=dndtnaDtq-nf(t)=1Γ(n-q)dndtn(∫at(t-s)n-q-1f(s)ds),tDbqf(t)=(-1)ndndtntDbq-nf(t)=1Γ(n-q)(-1)ndndtn(∫tb(s-t)n-q-1f(s)ds),
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 byAC([a,b],ℝN). ACk([a,b],ℝN)(k=1,…) is the space of functions f such that f∈Ck-1([a,b],ℝN) and f(k-1)∈AC([a,b],ℝN). In particular, AC([a,b],ℝN)=AC1([a,b],ℝN).Definition 2.3 (see [1]).
Letq≥0 and n∈ℕ.  If q∈[n-1,n) and f(t)∈ACn([a,b],ℝN), then the left and right Caputo fractional derivative of order q for function f(t) denoted by acDtqf(t) and tcDbqf(t), respectively, exist almost everywhere on [a,b].   acDtqf(t) and tcDbqf(t) are represented by
(2.3)acDtqf(t)=aDtq-nf(n)(t)=1Γ(n-q)(∫at(t-s)n-q-1f(n)(s)ds),(2.4)tcDbqf(t)=(-1)ntDbq-nf(n)(t)=(-1)nΓ(n-q)(∫tb(s-t)n-q-1f(n)(s)ds),
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,(2.5)aDt-q1(aDt-q2f(t))=aDt-q1-q2f(t),tDb-q1(tDb-q2f(t))=tDb-q1-q2f(t),∀q1,q2>0.Definition 2.5 (see [32]).
Define0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is defined by the closure of C0∞([0,T],ℝN) with respect to the norm
(2.6)‖u‖α,p=(∫0T|u(t)|pdt+∫0T|0cDtαu(t)|pdt)1/p,∀u∈E0α,p,
where C0∞([0,T],ℝN) denotes the set of all functions u∈C∞([0,T],ℝN) with u(0)=u(T)=0. It is obvious that the fractional derivative space E0α,p is the space of functions u∈Lp(0,T;ℝN) having an α-order Caputo fractional derivative 0cDtαu∈Lp(0,T;ℝN) and u(0)=u(T)=0.Proposition 2.6 (see [32]).
Let0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is a reflexive and separable space.Proposition 2.7 (see [32]).
Let0<α≤1 and 1<p<∞. For all u∈E0α,p, one has
(2.7)‖u‖Lp≤TαΓ(α+1)‖0cDtαu‖Lp.
Moreover, if α>1/p and 1/p+1/q=1, then
(2.8)‖u‖∞≤Tα-1/pΓ(α)((α-1)q+1)1/q‖0cDtαu‖Lp.According to (2.8), we can consider E0α,p with respect to the norm (2.9)‖u‖α,p=‖0cDtαu‖Lp=(∫0T|0cDtαu(t)|pdt)1/p.Proposition 2.8 (see [32]).
Define0<α≤1 and 1<p<∞. Assume that α>1/p and the sequence {uk} converges weakly to u in E0α,p, that is, uk⇀u. Then uk→u in C([0,T],ℝN), that is, ∥u-uk∥∞→0, as k→∞.Making use of Property2.4 and Definition 2.3, for any u∈AC([0,T],ℝN), BVP (1.1) is equivalent to the following problem: (2.10)ddt(120Dtα-1(0cDtαu(t))-12tDTα-1(tcDTαu(t)))+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,
where α=1-β/2∈(1/2,1].In the following, we will treat BVP (2.10) in the Hilbert space Eα=E0α,2 with the corresponding norm ∥u∥α=∥u∥α,2. The variational structure of BVP (2.10) on the space Eα has been established.Lemma 2.9 (see [32]).
LetL:[0,T]×ℝN×ℝN×ℝN→ℝ be defined by
(2.11)L(t,x,y,z)=-12(y,z)-F(t,x),
where F:[0,T]×ℝN→ℝ satisfies the assumption (A).If1/2<α≤1, then the functional defined by (2.12)φ(u)=∫0TL(t,u(t),0cDtαu(t),tcDTαu(t))dt
is continuously differentiable on Eα, and ∀u,v∈Eα, we have (2.13)〈φ′(u),v〉=∫0T(DxL(t,u(t),0cDtαu(t),tcDTαu(t)),v(t))dt+∫0T(DyL(t,u(t),0cDtαu(t),tcDTαu(t)),0cDtαv(t))dt+∫0T(DzL(t,u(t),0cDtαu(t),tcDTαu(t)),tcDTαv(t))dt.Definition 2.10 (see [32]).
A functionu∈AC([0,T],ℝN) is called a solution of BVP (2.10) if  (i)
Dα(u(t)) is derivative for almost every t∈[0,T],(ii)
u satisfies (2.10), where Dα(u(t)):=(1/2)0Dtα-1(0cDtαu(t))-(1/2)tDTα-1(tcDTαu(t)).Lemma 2.11 (see [32]).
Let1/2<α≤1 and φ be defined by (2.12). If assumption (A) is satisfied and u∈Eα 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 Lemma2.11, 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]).
If1/2<α≤1, then for any u∈Eα, one has
(2.14)|cos⁡(πα)|‖u‖α2≤-∫0T(0cDtαu(t),tcDTαu(t))dt≤1|cos⁡(πα)|‖u‖α2.Lemma 2.13 (see [23]).
LetX be a real Banach space, Φ:X→ℝ is differentiable. One says that Φ satisfies the (PS) condition if any sequence {uk} in X such that {Φ(uk)} is bounded and Φ′(uk)→0 as k→∞ contains a convergent subsequence.Lemma 2.14 (Mountain Pass theorem [24]).
LetX be a real Banach space and Φ:X→ℝ is differentiable and satisfies the (PS) condition. Suppose that  (i)
Φ(0)=0,(ii)
there existρ>0 and σ>0 such that Φ(z)≥σ for all z∈X with ∥z∥=ρ,(iii)
there existsz1 in X with ∥z1∥≥ρ such that Φ(z1)<σ.Then Φ possesses a critical value c≥σ. Moreover, c can be characterized as
(2.15)c=inf⁡g∈Ωmax⁡z∈g([0,1])Φ(z),
where Ω̅={g∈C([0,1],X):g(0)=0,g(1)=z1}.Lemma 2.15 (Clark theorem  [24]).
LetX 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 Sm-1, m∈ℕ, by an odd map, and supKΦ<0. Then Φ possesses at least m distinct pairs of critical points.


## 3. Proof of the Theorems

Foru∈Eα, where (3.1)Eα:={u∈L2(0,T;RN):0cDtαu∈L2(0,T;RN)}
is a reflexive Banach space with the norm defined by (3.2)‖u‖α=‖0cDtαu‖L2,‖u‖∞:=max⁡t∈[0,T]|u(t)|.It follows from Lemma2.9 that the functional φ on Eα given by (3.3)φ(u)=∫0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt
is continuously differentiable on Eα. Moreover, we have (3.4)〈φ′(u),v〉=-∫0T12[(0cDtαu(t),tcDTαv(t))+(tcDTαu(t),0cDtαv(t))]dt-∫0T(∇F(t,u(t)),v(t))dt.Recall that a sequence{un}∈Eα is said to be a (C) sequence of φ if φ(un) is bounded and (1+∥un∥α)∥φ(un)∥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 Theorem1.1

We will first establish the following lemma and then give the proof of Theorem1.1.Lemma 3.1.
Assume (A), (A2), and (A3) hold, then the functionalφ satisfies condition (C).Proof of Lemma3.1.
Let{un}⊂Eα be a (C) sequence of φ, that is, φ(un) is bounded and (1+∥un∥α)∥φ′(un)∥α→0 as n→∞. Then there exists M0 such that
(3.5)|φ(un)|≤M0,(1+‖un‖α)‖φ′(un)‖α≤M0,
for all n∈ℕ.
By (A2), there exist positive constantsB1 and M1 such that
(3.6)F(t,x)≤B1|x|r,
for all |x|≥M1 and a.e. t∈[0,T].
It follows from(A) that
(3.7)|F(t,x)|≤max⁡s∈[0,M1]a(s)b(t),
for all |x|≤M1 and a.e. t∈[0,T]. Therefore, we obtain
(3.8)F(t,x)≤B1|x|r+max⁡s∈[0,M1]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
Combining (2.14) and (3.8), we get
(3.9)|cos⁡(πα)|2‖un‖α2≤φ(un)+∫0TF(t,un(t))dt≤M0+max⁡s∈[0,M1]a(s)∫0Tb(t)dt+B1∫0T|un(t)|rdt.
On the other hand, by (A3), there existη>0 and M2>0 such that
(3.10)(∇F(t,x),x)-2F(t,x)≥η|x|μ,
for a.e. t∈[0,T] and |x|≥M2.
By (A), we have(3.11)|(∇F(t,x),x)-2F(t,x)|≤(2+M2)max⁡s∈[0,M2]a(s)b(t),
for all |x|≤M2 and a.e. t∈[0,T].
Therefore, we obtain(3.12)(∇F(t,x),x)-2F(t,x)≥η|x|μ-ηM2μ-(2+M2)max⁡s∈[0,M2]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
It follows from (3.5) and (3.12) that
(3.13)3M0≥2φ(un)-〈φ′(un),un〉=2∫0T[-12(0cDtαun(t),tcDTαun(t))-F(t,un(t))]dt-∫0T[-(0cDtαun(t),tcDTαun(t))-(∇F(t,un(t)),un(t))]dt=∫0T[(∇F(t,un(t)),un(t))-2F(t,un(t))]dt≥η∫0T|un(t)|μdt-(2+M2)max⁡s∈[0,M2]a(s)∫0Tb(t)dt-ηM2μT,
thus, ∫0T|un(t)|μdt is bounded.
Ifμ>r, then
(3.14)∫0T|un(t)|rdt≤T(μ-r)/μ(∫0T|un(t)|μdt)r/μ,
which, combining (3.9), implies that ∥un∥α is bounded.
Ifμ≤r, then
(3.15)∫0T|un(t)|rdt≤‖un‖∞r-μ∫0T|un(t)|μdt≤C1r-μ‖un‖αr-μ∫0T|un(t)|μdt,
where
(3.16)C1:=Tα-1/2Γ(α)(2α-1)1/2,
by (2.8).
Sinceμ>r-2, it follows from (3.9) that ∥un∥α is bounded too. Thus ∥un∥α is bounded in Eα.
By Proposition2.8, the sequence {un} has a subsequence, also denoted by {un}, such that
(3.17)un⇀uweaklyinEα,un⟶ustronglyinC([0,T],RN).
Then we obtainun→u in Eα by use of the same argument of Theorem  5.2 in [32]. The proof of Lemma 3.1 is completed.Proof of Theorem1.1.
By (A1), there existϵ1∈(0,|cos(πα)|) and δ>0 such that
(3.18)F(t,x)≤(|cos⁡(πα)|-ϵ1)Γ2(α+1)2T2α|x|2,
for a.e. t∈[0,T] and x∈ℝN with |x|≤δ.
Let(3.19)ρ=Γ(α)(2(α-1)+1)1/2Tα-1/2δ,σ=ϵ1ρ22>0.
Then it follows from (2.8) that
(3.20)‖u‖∞≤Tα-1/2Γ(α)(2(α-1)+1)1/2‖u‖α=δ,
for all u∈Eα with ∥u∥α=ρ.
Therefore, we have(3.21)φ(u)=∫0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt≥|cos⁡(πα)|2∥u∥α2-(|cos⁡(πα)|-ϵ1)Γ2(α+1)2T2α∫0T|u(t)|2dt≥|cos⁡(πα)|2‖u‖α2-|cos⁡(πα)|-ϵ12‖u‖α2=ϵ12‖u‖α2=σ,
for all u∈Eα with ∥u∥α=ρ. This implies that (ii) in Lemma 2.14 is satisfied.
It is obvious from the definition ofφ and (A1) that φ(0)=0, and therefore, it suffices to show that φ satisfies (iii) in Lemma 2.14.
By (A1), there existϵ2>0 and M3>0 such that
(3.22)F(t,x)>(π2|cos⁡(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)|x|2,
for all |x|≥M3 and a.e. t∈[0,T].
It follows from (A) that(3.23)|F(t,x)|≤max⁡s∈[0,M3]a(s)b(t),
for all |x|≤M3 and a.e. t∈[0,T].
Therefore, we obtain(3.24)F(t,x)≥(π2|cos⁡(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)(|x|2-M32)-max⁡s∈[0,M3]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
Choosingu0=((T/π)sin(πt/T),0,…,0)∈Eα, then
(3.25)‖u0‖L22=T32π2,‖u0‖α2≤T3-2αΓ2(2-α)(3-2α).
Forς>0 and noting that (3.24) and (3.25), we have
(3.26)φ(ςu0)=∫0T[-12(0cDtαςu0(t),tcDTαςu0(t))-F(t,ςu0(t))]dt≤ς22|cos⁡(πα)|‖u0‖α2-(ς2π2|cos⁡(πα)|T2αΓ2(2-α)(3-2α)+ς2ϵ2)∫0T|u0(t)|2dt+C2≤ς22|cos⁡(πα)|⋅T3-2αΓ2(2-α)(3-2α)-ς2π2|cos⁡(πα)|T2αΓ2(2-α)(3-2α)⋅T32π2-ς2ϵ2T32π2+C2⟶-∞,
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 thatφ(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 Theorem1.2

The following lemmata are needed in the proof of Theorem1.2.Lemma 3.2.
Assume (A5), then for anyε>0, there exists a subset Eε⊂[0,T] with meas([0,T]∖Eε)<ε such that
(3.27)lim⁡|x|→∞[(∇F(t,x),x)-2F(t,x)]=+∞,
uniformly for t∈Eε.Proof of Lemma3.2.
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 functionalφ satisfies condition (C).Proof of Lemma3.3.
Suppose that{un}⊂Eα is a (C) sequence of φ, that is, φ(un) is bounded and (1+∥un∥α)∥φ′(un)∥α→0 as n→∞. Then we have
(3.28)liminf⁡n→∞[〈φ′(un),un〉-2φ(un)]>-∞,
which implies that
(3.29)limsup⁡n→∞∫0T[(∇F(t,un),un)-2F(t,un)]dt<+∞.
We only need to show that{un} is bounded in Eα. If {un} is unbounded, we may assume, without loss of generality, that ∥un∥α→∞ as n→∞. Put zn=un/∥un∥α, we then have ∥zn∥α=1. Going to a sequence if necessary, we assume that zn⇀z weakly in Eα, zn→z strongly in C([0,T],ℝN) and L2(0,T;ℝN).
By (A2), it follows that there exist constantsB2>0 and M4>0 such that
(3.30)F(t,x)≤B2|x|2,
for all |x|≥M4 and a.e. t∈[0,T].
By assumption (A), it follows that(3.31)|F(t,x)|≤max⁡s∈[0,M4]a(s)b(t),
for all |x|≤M4 and a.e. t∈[0,T]. Therefore, we obtain
(3.32)F(t,x)≤B2|x|2+max⁡s∈[0,M4]a(s)b(t)
for all x∈ℝN and a.e. t∈[0,T]. Therefore, we have
(3.33)φ(u)=∫0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt≥|cos⁡(πα)|2‖u‖α2-B2∫0T|u|2dt-max⁡s∈[0,M4]a(s)∫0Tb(t)dt,
from which, it follows that
(3.34)φ(un)‖un‖α2≥|cos⁡(πα)|2-B2‖zn‖L22-1‖un‖α2max⁡s∈[0,M4]a(s)∫0Tb(t)dt.
Passing to the limit in the last inequality, we get(3.35)|cos⁡(πα)|2-B2‖z‖L22≤0,
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 Lemma3.2, for ε=(1/2) meas(E)>0, we can choose a subset Eε⊂[0,T] with meas([0,T]∖Eε)<ε such that
(3.36)lim⁡|x|→∞[(∇F(t,x),x)-2F(t,x)]=+∞,
uniformly for t∈Eε.
We assert that meas(E⋂Eε)>0. If not, meas(E⋂Eε)=0.
SinceE=(E⋂Eε)⋃(E∖Eε), it follows that
(3.37)0<meas⁡(E)=meas⁡(E∩Eε)+meas⁡(E∖Eε)≤meas⁡([0,T]∖Eε)<ε=12meas⁡(E),
which leads to a contradiction and establishes the assertion.
By (A4), we obtain thye following:(3.38)∫0T[(∇F(t,un),un)-2F(t,un)]dt=∫E⋂Eε[(∇F(t,un),un)-2F(t,un)]dt+∫[0,T]∖(E⋂Eε)[(∇F(t,un),un)-2F(t,un)]dt≥∫E⋂Eε[(∇F(t,un),un)-2F(t,un)]dt-∫0T|τ(t)|dt.
By (3.36), (3.38), and Fatou's lemma, it follows that
(3.39)lim⁡n→∞∫0T[(∇F(t,un),un)-2F(t,un)]dt=+∞,
which contradicts (3.29). This contradiction shows that ∥un∥α is bounded in Eα, and this completes the proof.
By virtue of Lemmas3.2 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 Theorem1.4

The proof of Theorem1.4 is divided into a sequence of lemma.Lemma 3.4.
The functionalφ is bounded below on Eα.Proof of Lemma3.4.
By (2.8) and (2.14), for every u∈Eα, we have
(3.40)φ(u)=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0TF(t,u(t))dt=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0Ta(t)|u(t)|γdt≥|cos⁡(πα)|2‖u‖α2-a0‖u‖∞γT≥|cos⁡(πα)|2‖u‖α2-a0TC1γ‖u‖αγ,
where a0=esssup{a(t):t∈[0,T]}. The proof of Lemma 3.4 is complete.Lemma 3.5.
The functionalφ satisfies the (PS) condition.Proof of Lemma3.5.
Let{un} be a Palais-Smale sequence in Eα, that is,
(3.41)φ(un)isboundedandφ′(un)⟶0asn⟶+∞.
Suppose that{un} is unbounded in Eα, that is, ∥un∥α→+∞ as n→+∞. Since
(3.42)〈φ′(un),un〉-γφ(un)=(-1+γ2)∫0T(0cDtαun(t),tcDTαun(t))dt.
However, from (3.42), we have
(3.43)-γφ(un)≥(1-γ2)|cos⁡(πα)|‖un‖α2-‖φ′(un)‖‖un‖α,
thus ∥un∥α is a bounded sequence in Eα. Since Eα is a reflexive space, going, if necessary, to a subsequence, we can assume that un⇀u in Eα, thus we have
(3.44)〈φ′(un)-φ′(u),un-u〉=〈φ′(un),un-u〉-〈φ′(u),un-u〉≤‖φ′(un)‖α‖un-u‖α-〈φ′(u),un-u〉⟶0,
as n→∞. Moreover, according to (2.8) and Proposition 2.8, we have that {un} is bounded in C([0,T],ℝN) and ∥un-u∥∞→0 as n→∞.
Noting that(3.45)〈φ′(un)-φ′(u),un-u〉=-∫0T(0cDtα(un(t)-u(t)),0cDTα(un(t)-u(t)))dt-∫0T(∇F(t,un(t))-∇F(t,u(t)),un(t)-u(t))dt≥|cos⁡(πα)|‖un-u‖α2-|∫0T(∇F(t,un(t))-∇F(t,u(t)))dt|‖un-u‖∞.
Combining (3.44) and (3.45), it is easy to verify that ∥un-u∥α→0 as n→∞, and hence that un→u in Eα. Thus, {un} admits a convergent subsequence. The proof of Lemma 3.5 is complete.Lemma 3.6.
For anym∈ℕ, there exists a set K⊂Eα which is homeomorphic to Sm-1 by an odd map, and supkφ<0.Proof of Lemma3.6.
For everym∈ℕ, define
(3.46)ui(t)=(sin⁡⁡iπtT,0,…,0),i=1,2,…,m,Em=span⁡⁡{u1,…,um},Km,β={u∈Em:‖u‖α=β},
where β is a positive number to be chosen later.
For anyu∈Em, there exist λi∈ℝ, i=1,2,…,m, such that
(3.47)u=∑i=1mλiui(t),‖u‖α2=∫0T|0cDtαu(t)|2dt=∫0T(0cDtαu(t),0cDtαu(t))dt=∫0T(λ10cDtαu1(t)+⋯+λm0cDtαum(t),λ10cDtαu1(t)+⋯+λm0cDtαum(t))dt=∑i=1m∑j=1maijλiλj=F(λ1,…,λm),
where aij=∫0T(0cDtαui(t),0cDtαuj(t))dt and F(λ1,…,λm) is a real quadratic form.
Since(3.48)F(λ1,…,λm)=‖∑i=1mλiui(t)‖α2≥0,∀(λ1,…,λm)T∈Rm,F(λ1,…,λm)=0⟺∑i=1mλiui(t)≡0⟺λ1=λ2=⋯=λm=0.
So, F(λ1,…,λm) is a real positive definite quadratic form. Then there exist an invertible matrix C∈ℝm×m and μi∈ℝ, i=1,2,…,m, such that
(3.49)(λ1,λ2,…,λm)T=C(μ1,μ2,…,μm)T,F(λ1,…,λm)=∑i=1mμi2.
It is easy to prove that the odd mappingΨ:Km,β→Sm-1 defined by
(3.50)Ψ(u)=β-1(μ1,…,μm)
is a homeomorphism between Km,β and Sm-1.
SinceEm⊂Eα is a finite dimensional space, there exists ε(m)>0 such that
(3.51)meas⁡{t∈[0,T]:a(t)|u(t)|γ≥ε‖u‖αγ}≥ε,∀u∈Em∖{0}.
Otherwise, for any positive integern, there exists un∈Em∖{0} such that
(3.52)meas⁡{t∈[0,T]:a(t)|un(t)|γ≥1n‖un‖αγ}<1n.
Setvn(t):=un(t)/∥un∥α∈Em∖{0}, then ∥vn∥α=1 for all n∈ℕ and
(3.53)meas⁡{t∈[0,T]:a(t)|vn(t)|γ≥1n}<1n.
SincedimEm<∞, it follows from the compactness of the unit sphere of Em that there exists a subsequence, denoted also by {vn}, such that {vn} converges to some v0 in Em. It is obvious that ∥v0∥α=1.
By the equivalence of the norms on the finite dimensional space, we havevn→v0 in L2(0,T;ℝN), that is,
(3.54)∫0T|vn-v0|2dt⟶0asn⟶∞.
By (3.54) and Hölder inequality, we have
(3.55)∫0Ta(t)|vn-v0|γdt≤(∫0Ta(t)2/(2-γ)dt)(2-γ)/2(∫0T|vn-v0|2dt)γ/2=‖a‖(2-γ)/2(∫0T|vn-v0|2dt)γ/2⟶0,asn⟶∞.
Thus, there existξ1,ξ2>0 such that
(3.56)meas⁡{t∈[0,T]:a(t)|v0(t)|γ≥ξ1}≥ξ2.
In fact, if not, we have(3.57)meas⁡{t∈[0,T]:a(t)|v0(t)|γ≥1n}=0,
for all positive integer n.
It implies that(3.58)0≤∫0Ta(t)|v0|γ+2dt<Tn‖v0‖∞2≤C12Tn‖v0‖α2⟶0,
as n→∞. Hence v0=0 which contradicts that ∥v0∥α=1. Therefore, (3.56) holds.
Now let(3.59)Ω0={t∈[0,T]:a(t)|v0(t)|γ≥ξ1},Ωn={t∈[0,T]:a(t)|vn(t)|γ<1n},
and Ωnc=[0,T]∖Ωn={t∈[0,T]:a(t)|vn(t)|γ≥1/n}.
By (3.53) and (3.56), we have
(3.60)meas⁡(Ωn∩Ω0)=meas⁡(Ω0∖(Ωnc∩Ω0))≥meas⁡(Ω0)-meas⁡(Ωnc∩Ω0)≥ξ2-1n,
for all positive integer n. Let n be large enough such that
(3.61)ξ2-1n≥12ξ2,12γ-1ξ1-1n≥12γξ1,
then we have
(3.62)∫0Ta(t)|vn-v0|γdt≥∫Ωn∩Ω0a(t)|vn-v0|γdt≥12γ-1∫Ωn∩Ω0a(t)|v0|γdt-∫Ωn∩Ω0a(t)|vn|γdt≥(12γ-1ξ1-1n)meas⁡(Ωn∩Ω0)≥ξ12γ⋅ξ22=ξ1ξ22γ+1>0,
for all large n, which is a contradiction to (3.55). Therefore, (3.51) holds.
For anyu∈Km,β, we have
(3.63)φ(u)=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0TF(t,u(t))dt≤12|cos⁡(πα)|‖u‖α2-∫0Ta(t)|u(t)|γdt≤12|cos⁡(πα)|‖u‖α2-ε‖u‖αγmeas⁡(Ωu)≤12|cos⁡(πα)|‖u‖α2-ε2‖u‖αγ,
by (3.51), where Ωu:={t∈[0,T]:a(t)|u(t)|γ≥ε∥u∥αγ}.
Choosingβ=(|cos(πα)|ε2)1/(2-γ), we conclude supKm,βφ<-ε2βγ/2<0 which completes the proof.Now from the assertion of Lemma2.15, 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 Eα. The proof of Theorem 1.4 is completed.


## 3.1. Proof of Theorem1.1

We will first establish the following lemma and then give the proof of Theorem1.1.Lemma 3.1.
Assume (A), (A2), and (A3) hold, then the functionalφ satisfies condition (C).Proof of Lemma3.1.
Let{un}⊂Eα be a (C) sequence of φ, that is, φ(un) is bounded and (1+∥un∥α)∥φ′(un)∥α→0 as n→∞. Then there exists M0 such that
(3.5)|φ(un)|≤M0,(1+‖un‖α)‖φ′(un)‖α≤M0,
for all n∈ℕ.
By (A2), there exist positive constantsB1 and M1 such that
(3.6)F(t,x)≤B1|x|r,
for all |x|≥M1 and a.e. t∈[0,T].
It follows from(A) that
(3.7)|F(t,x)|≤max⁡s∈[0,M1]a(s)b(t),
for all |x|≤M1 and a.e. t∈[0,T]. Therefore, we obtain
(3.8)F(t,x)≤B1|x|r+max⁡s∈[0,M1]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
Combining (2.14) and (3.8), we get
(3.9)|cos⁡(πα)|2‖un‖α2≤φ(un)+∫0TF(t,un(t))dt≤M0+max⁡s∈[0,M1]a(s)∫0Tb(t)dt+B1∫0T|un(t)|rdt.
On the other hand, by (A3), there existη>0 and M2>0 such that
(3.10)(∇F(t,x),x)-2F(t,x)≥η|x|μ,
for a.e. t∈[0,T] and |x|≥M2.
By (A), we have(3.11)|(∇F(t,x),x)-2F(t,x)|≤(2+M2)max⁡s∈[0,M2]a(s)b(t),
for all |x|≤M2 and a.e. t∈[0,T].
Therefore, we obtain(3.12)(∇F(t,x),x)-2F(t,x)≥η|x|μ-ηM2μ-(2+M2)max⁡s∈[0,M2]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
It follows from (3.5) and (3.12) that
(3.13)3M0≥2φ(un)-〈φ′(un),un〉=2∫0T[-12(0cDtαun(t),tcDTαun(t))-F(t,un(t))]dt-∫0T[-(0cDtαun(t),tcDTαun(t))-(∇F(t,un(t)),un(t))]dt=∫0T[(∇F(t,un(t)),un(t))-2F(t,un(t))]dt≥η∫0T|un(t)|μdt-(2+M2)max⁡s∈[0,M2]a(s)∫0Tb(t)dt-ηM2μT,
thus, ∫0T|un(t)|μdt is bounded.
Ifμ>r, then
(3.14)∫0T|un(t)|rdt≤T(μ-r)/μ(∫0T|un(t)|μdt)r/μ,
which, combining (3.9), implies that ∥un∥α is bounded.
Ifμ≤r, then
(3.15)∫0T|un(t)|rdt≤‖un‖∞r-μ∫0T|un(t)|μdt≤C1r-μ‖un‖αr-μ∫0T|un(t)|μdt,
where
(3.16)C1:=Tα-1/2Γ(α)(2α-1)1/2,
by (2.8).
Sinceμ>r-2, it follows from (3.9) that ∥un∥α is bounded too. Thus ∥un∥α is bounded in Eα.
By Proposition2.8, the sequence {un} has a subsequence, also denoted by {un}, such that
(3.17)un⇀uweaklyinEα,un⟶ustronglyinC([0,T],RN).
Then we obtainun→u in Eα by use of the same argument of Theorem  5.2 in [32]. The proof of Lemma 3.1 is completed.Proof of Theorem1.1.
By (A1), there existϵ1∈(0,|cos(πα)|) and δ>0 such that
(3.18)F(t,x)≤(|cos⁡(πα)|-ϵ1)Γ2(α+1)2T2α|x|2,
for a.e. t∈[0,T] and x∈ℝN with |x|≤δ.
Let(3.19)ρ=Γ(α)(2(α-1)+1)1/2Tα-1/2δ,σ=ϵ1ρ22>0.
Then it follows from (2.8) that
(3.20)‖u‖∞≤Tα-1/2Γ(α)(2(α-1)+1)1/2‖u‖α=δ,
for all u∈Eα with ∥u∥α=ρ.
Therefore, we have(3.21)φ(u)=∫0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt≥|cos⁡(πα)|2∥u∥α2-(|cos⁡(πα)|-ϵ1)Γ2(α+1)2T2α∫0T|u(t)|2dt≥|cos⁡(πα)|2‖u‖α2-|cos⁡(πα)|-ϵ12‖u‖α2=ϵ12‖u‖α2=σ,
for all u∈Eα with ∥u∥α=ρ. This implies that (ii) in Lemma 2.14 is satisfied.
It is obvious from the definition ofφ and (A1) that φ(0)=0, and therefore, it suffices to show that φ satisfies (iii) in Lemma 2.14.
By (A1), there existϵ2>0 and M3>0 such that
(3.22)F(t,x)>(π2|cos⁡(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)|x|2,
for all |x|≥M3 and a.e. t∈[0,T].
It follows from (A) that(3.23)|F(t,x)|≤max⁡s∈[0,M3]a(s)b(t),
for all |x|≤M3 and a.e. t∈[0,T].
Therefore, we obtain(3.24)F(t,x)≥(π2|cos⁡(πα)|Γ2(2-α)T2α(3-2α)+ϵ2)(|x|2-M32)-max⁡s∈[0,M3]a(s)b(t),
for all x∈ℝN and a.e. t∈[0,T].
Choosingu0=((T/π)sin(πt/T),0,…,0)∈Eα, then
(3.25)‖u0‖L22=T32π2,‖u0‖α2≤T3-2αΓ2(2-α)(3-2α).
Forς>0 and noting that (3.24) and (3.25), we have
(3.26)φ(ςu0)=∫0T[-12(0cDtαςu0(t),tcDTαςu0(t))-F(t,ςu0(t))]dt≤ς22|cos⁡(πα)|‖u0‖α2-(ς2π2|cos⁡(πα)|T2αΓ2(2-α)(3-2α)+ς2ϵ2)∫0T|u0(t)|2dt+C2≤ς22|cos⁡(πα)|⋅T3-2αΓ2(2-α)(3-2α)-ς2π2|cos⁡(πα)|T2αΓ2(2-α)(3-2α)⋅T32π2-ς2ϵ2T32π2+C2⟶-∞,
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 thatφ(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 Theorem1.2

The following lemmata are needed in the proof of Theorem1.2.Lemma 3.2.
Assume (A5), then for anyε>0, there exists a subset Eε⊂[0,T] with meas([0,T]∖Eε)<ε such that
(3.27)lim⁡|x|→∞[(∇F(t,x),x)-2F(t,x)]=+∞,
uniformly for t∈Eε.Proof of Lemma3.2.
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 functionalφ satisfies condition (C).Proof of Lemma3.3.
Suppose that{un}⊂Eα is a (C) sequence of φ, that is, φ(un) is bounded and (1+∥un∥α)∥φ′(un)∥α→0 as n→∞. Then we have
(3.28)liminf⁡n→∞[〈φ′(un),un〉-2φ(un)]>-∞,
which implies that
(3.29)limsup⁡n→∞∫0T[(∇F(t,un),un)-2F(t,un)]dt<+∞.
We only need to show that{un} is bounded in Eα. If {un} is unbounded, we may assume, without loss of generality, that ∥un∥α→∞ as n→∞. Put zn=un/∥un∥α, we then have ∥zn∥α=1. Going to a sequence if necessary, we assume that zn⇀z weakly in Eα, zn→z strongly in C([0,T],ℝN) and L2(0,T;ℝN).
By (A2), it follows that there exist constantsB2>0 and M4>0 such that
(3.30)F(t,x)≤B2|x|2,
for all |x|≥M4 and a.e. t∈[0,T].
By assumption (A), it follows that(3.31)|F(t,x)|≤max⁡s∈[0,M4]a(s)b(t),
for all |x|≤M4 and a.e. t∈[0,T]. Therefore, we obtain
(3.32)F(t,x)≤B2|x|2+max⁡s∈[0,M4]a(s)b(t)
for all x∈ℝN and a.e. t∈[0,T]. Therefore, we have
(3.33)φ(u)=∫0T[-12(0cDtαu(t),tcDTαu(t))-F(t,u(t))]dt≥|cos⁡(πα)|2‖u‖α2-B2∫0T|u|2dt-max⁡s∈[0,M4]a(s)∫0Tb(t)dt,
from which, it follows that
(3.34)φ(un)‖un‖α2≥|cos⁡(πα)|2-B2‖zn‖L22-1‖un‖α2max⁡s∈[0,M4]a(s)∫0Tb(t)dt.
Passing to the limit in the last inequality, we get(3.35)|cos⁡(πα)|2-B2‖z‖L22≤0,
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 Lemma3.2, for ε=(1/2) meas(E)>0, we can choose a subset Eε⊂[0,T] with meas([0,T]∖Eε)<ε such that
(3.36)lim⁡|x|→∞[(∇F(t,x),x)-2F(t,x)]=+∞,
uniformly for t∈Eε.
We assert that meas(E⋂Eε)>0. If not, meas(E⋂Eε)=0.
SinceE=(E⋂Eε)⋃(E∖Eε), it follows that
(3.37)0<meas⁡(E)=meas⁡(E∩Eε)+meas⁡(E∖Eε)≤meas⁡([0,T]∖Eε)<ε=12meas⁡(E),
which leads to a contradiction and establishes the assertion.
By (A4), we obtain thye following:(3.38)∫0T[(∇F(t,un),un)-2F(t,un)]dt=∫E⋂Eε[(∇F(t,un),un)-2F(t,un)]dt+∫[0,T]∖(E⋂Eε)[(∇F(t,un),un)-2F(t,un)]dt≥∫E⋂Eε[(∇F(t,un),un)-2F(t,un)]dt-∫0T|τ(t)|dt.
By (3.36), (3.38), and Fatou's lemma, it follows that
(3.39)lim⁡n→∞∫0T[(∇F(t,un),un)-2F(t,un)]dt=+∞,
which contradicts (3.29). This contradiction shows that ∥un∥α is bounded in Eα, and this completes the proof.
By virtue of Lemmas3.2 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 Theorem1.4

The proof of Theorem1.4 is divided into a sequence of lemma.Lemma 3.4.
The functionalφ is bounded below on Eα.Proof of Lemma3.4.
By (2.8) and (2.14), for every u∈Eα, we have
(3.40)φ(u)=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0TF(t,u(t))dt=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0Ta(t)|u(t)|γdt≥|cos⁡(πα)|2‖u‖α2-a0‖u‖∞γT≥|cos⁡(πα)|2‖u‖α2-a0TC1γ‖u‖αγ,
where a0=esssup{a(t):t∈[0,T]}. The proof of Lemma 3.4 is complete.Lemma 3.5.
The functionalφ satisfies the (PS) condition.Proof of Lemma3.5.
Let{un} be a Palais-Smale sequence in Eα, that is,
(3.41)φ(un)isboundedandφ′(un)⟶0asn⟶+∞.
Suppose that{un} is unbounded in Eα, that is, ∥un∥α→+∞ as n→+∞. Since
(3.42)〈φ′(un),un〉-γφ(un)=(-1+γ2)∫0T(0cDtαun(t),tcDTαun(t))dt.
However, from (3.42), we have
(3.43)-γφ(un)≥(1-γ2)|cos⁡(πα)|‖un‖α2-‖φ′(un)‖‖un‖α,
thus ∥un∥α is a bounded sequence in Eα. Since Eα is a reflexive space, going, if necessary, to a subsequence, we can assume that un⇀u in Eα, thus we have
(3.44)〈φ′(un)-φ′(u),un-u〉=〈φ′(un),un-u〉-〈φ′(u),un-u〉≤‖φ′(un)‖α‖un-u‖α-〈φ′(u),un-u〉⟶0,
as n→∞. Moreover, according to (2.8) and Proposition 2.8, we have that {un} is bounded in C([0,T],ℝN) and ∥un-u∥∞→0 as n→∞.
Noting that(3.45)〈φ′(un)-φ′(u),un-u〉=-∫0T(0cDtα(un(t)-u(t)),0cDTα(un(t)-u(t)))dt-∫0T(∇F(t,un(t))-∇F(t,u(t)),un(t)-u(t))dt≥|cos⁡(πα)|‖un-u‖α2-|∫0T(∇F(t,un(t))-∇F(t,u(t)))dt|‖un-u‖∞.
Combining (3.44) and (3.45), it is easy to verify that ∥un-u∥α→0 as n→∞, and hence that un→u in Eα. Thus, {un} admits a convergent subsequence. The proof of Lemma 3.5 is complete.Lemma 3.6.
For anym∈ℕ, there exists a set K⊂Eα which is homeomorphic to Sm-1 by an odd map, and supkφ<0.Proof of Lemma3.6.
For everym∈ℕ, define
(3.46)ui(t)=(sin⁡⁡iπtT,0,…,0),i=1,2,…,m,Em=span⁡⁡{u1,…,um},Km,β={u∈Em:‖u‖α=β},
where β is a positive number to be chosen later.
For anyu∈Em, there exist λi∈ℝ, i=1,2,…,m, such that
(3.47)u=∑i=1mλiui(t),‖u‖α2=∫0T|0cDtαu(t)|2dt=∫0T(0cDtαu(t),0cDtαu(t))dt=∫0T(λ10cDtαu1(t)+⋯+λm0cDtαum(t),λ10cDtαu1(t)+⋯+λm0cDtαum(t))dt=∑i=1m∑j=1maijλiλj=F(λ1,…,λm),
where aij=∫0T(0cDtαui(t),0cDtαuj(t))dt and F(λ1,…,λm) is a real quadratic form.
Since(3.48)F(λ1,…,λm)=‖∑i=1mλiui(t)‖α2≥0,∀(λ1,…,λm)T∈Rm,F(λ1,…,λm)=0⟺∑i=1mλiui(t)≡0⟺λ1=λ2=⋯=λm=0.
So, F(λ1,…,λm) is a real positive definite quadratic form. Then there exist an invertible matrix C∈ℝm×m and μi∈ℝ, i=1,2,…,m, such that
(3.49)(λ1,λ2,…,λm)T=C(μ1,μ2,…,μm)T,F(λ1,…,λm)=∑i=1mμi2.
It is easy to prove that the odd mappingΨ:Km,β→Sm-1 defined by
(3.50)Ψ(u)=β-1(μ1,…,μm)
is a homeomorphism between Km,β and Sm-1.
SinceEm⊂Eα is a finite dimensional space, there exists ε(m)>0 such that
(3.51)meas⁡{t∈[0,T]:a(t)|u(t)|γ≥ε‖u‖αγ}≥ε,∀u∈Em∖{0}.
Otherwise, for any positive integern, there exists un∈Em∖{0} such that
(3.52)meas⁡{t∈[0,T]:a(t)|un(t)|γ≥1n‖un‖αγ}<1n.
Setvn(t):=un(t)/∥un∥α∈Em∖{0}, then ∥vn∥α=1 for all n∈ℕ and
(3.53)meas⁡{t∈[0,T]:a(t)|vn(t)|γ≥1n}<1n.
SincedimEm<∞, it follows from the compactness of the unit sphere of Em that there exists a subsequence, denoted also by {vn}, such that {vn} converges to some v0 in Em. It is obvious that ∥v0∥α=1.
By the equivalence of the norms on the finite dimensional space, we havevn→v0 in L2(0,T;ℝN), that is,
(3.54)∫0T|vn-v0|2dt⟶0asn⟶∞.
By (3.54) and Hölder inequality, we have
(3.55)∫0Ta(t)|vn-v0|γdt≤(∫0Ta(t)2/(2-γ)dt)(2-γ)/2(∫0T|vn-v0|2dt)γ/2=‖a‖(2-γ)/2(∫0T|vn-v0|2dt)γ/2⟶0,asn⟶∞.
Thus, there existξ1,ξ2>0 such that
(3.56)meas⁡{t∈[0,T]:a(t)|v0(t)|γ≥ξ1}≥ξ2.
In fact, if not, we have(3.57)meas⁡{t∈[0,T]:a(t)|v0(t)|γ≥1n}=0,
for all positive integer n.
It implies that(3.58)0≤∫0Ta(t)|v0|γ+2dt<Tn‖v0‖∞2≤C12Tn‖v0‖α2⟶0,
as n→∞. Hence v0=0 which contradicts that ∥v0∥α=1. Therefore, (3.56) holds.
Now let(3.59)Ω0={t∈[0,T]:a(t)|v0(t)|γ≥ξ1},Ωn={t∈[0,T]:a(t)|vn(t)|γ<1n},
and Ωnc=[0,T]∖Ωn={t∈[0,T]:a(t)|vn(t)|γ≥1/n}.
By (3.53) and (3.56), we have
(3.60)meas⁡(Ωn∩Ω0)=meas⁡(Ω0∖(Ωnc∩Ω0))≥meas⁡(Ω0)-meas⁡(Ωnc∩Ω0)≥ξ2-1n,
for all positive integer n. Let n be large enough such that
(3.61)ξ2-1n≥12ξ2,12γ-1ξ1-1n≥12γξ1,
then we have
(3.62)∫0Ta(t)|vn-v0|γdt≥∫Ωn∩Ω0a(t)|vn-v0|γdt≥12γ-1∫Ωn∩Ω0a(t)|v0|γdt-∫Ωn∩Ω0a(t)|vn|γdt≥(12γ-1ξ1-1n)meas⁡(Ωn∩Ω0)≥ξ12γ⋅ξ22=ξ1ξ22γ+1>0,
for all large n, which is a contradiction to (3.55). Therefore, (3.51) holds.
For anyu∈Km,β, we have
(3.63)φ(u)=-∫0T12(0cDtαu(t),tcDTαu(t))dt-∫0TF(t,u(t))dt≤12|cos⁡(πα)|‖u‖α2-∫0Ta(t)|u(t)|γdt≤12|cos⁡(πα)|‖u‖α2-ε‖u‖αγmeas⁡(Ωu)≤12|cos⁡(πα)|‖u‖α2-ε2‖u‖αγ,
by (3.51), where Ωu:={t∈[0,T]:a(t)|u(t)|γ≥ε∥u∥αγ}.
Choosingβ=(|cos(πα)|ε2)1/(2-γ), we conclude supKm,βφ<-ε2βγ/2<0 which completes the proof.Now from the assertion of Lemma2.15, 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 Eα. 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
(4.1)F(t,x)=ln⁡(1+2|x|2)|x|2.
These show that all conditions of Theorem1.1 are satisfied, where
(4.2)r=2.5,μ=2.
By Theorem1.1, BVP (1.1) has at least one solution u∈Eα.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.
Letf(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
(4.3)κ>(2π)1-2α|cos⁡(πα)|Γ2(2-α)(3-2α).
Furthermore, we have
(4.4)(∇f(x),x)-2f(x)=2|x|21+|x|2-2ln⁡(1+|x|2)⟶-∞,
as |x|→+∞. Therefore, we have
(4.5)(∇F(t,x),x)-2F(t,x)=κ2|x|21+|x|4f(x)(2+sin⁡t)+κ[(∇f(x),x)-2f(x)](2+sin⁡t)arctan|x|2⟶-∞,
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 Eα.
Iff(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
(4.6)a(t)={T,t=02t,0<t≤T2-2(t-T),T2<t<TT,t=T.
By Theorem1.4, BVP (1.1) has infinite solutions on Eα.



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*Source: 648635-2012-01-18.xml*