# Infinitely Many Solutions for a Class of Fractional Boundary Value Problems with Nonsmooth Potential

Authors: Kaimin Teng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish the existence of infinitely many solutions for a class of fractional boundary value problems with nonsmooth potential. The
technical approach is mainly based on a result of infinitely many critical
points for locally Lipschitz functions.


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## Body

## 1. Introduction

In the present paper, we are concerned with the existence of infinitely many solutions for a class of fractional boundary value problems with the following form:(1)otDTα(o0Dtαu(t))∈λ∂F(t,u(t))a.e.t∈[0,T],hhhhhhhhhlhhu(0)=u(T)=0,
where λ∈ℝ is a parameter, and tDTα and 0Dtα are the left and right Riemann-Liouville fractional derivatives of order 0<α≤1, respectively. F:[0,T]×ℝN→ℝ is a given function satisfying some assumptions, and ∂F(t,·) is the generalized gradient in the sense of Clarke [1].In particular, ifα=1, then problem (1) is reduced to the standard second-order boundary value problem
(2)u¨(t)∈λ∂F(t,u(t))a.e.t∈[0,T],hhhhhhhhhhh.u(0)=u(T)=0.
There are many excellent results that have been worked out on the existence of solutions for second-order BVP (we refer the reader to see [2, 3] and the references therein).Recently, fractional differential equations and inclusions have attracted lots of people’s interests because of their applications in viscoelasticity, electrochemistry, control, porous media, and so forth. The existence and multiplicity of solutions for BVP of fractional differential equations and inclusions have been established by some fixed-point theorems; we refer the readers to see [4–7].This paper is motivated by the recent papers [8] where several existence results concerning problem (1) under the smooth case are obtained by using variational methods. In their papers, the authors define a suitable space and find a variational functional for fractional differential equations with Dirichlet boundary conditions. The aim of the present paper is to establish the existence of infinitely many solutions for problem (1) by using a critical points theorem according to Bonanno and Bisci [9].It is interesting that the existence of infinitely many solutions for differential equations can be established without the symmetry assumption. Recently, Bonanno and Bisci in [9] established a precise version of the infinitely many critical points theorem of Marano and Motreanu [10] which extended the results of Ricceri in [11] for the nondifferentiable functionals. In applying the theorem, we need to assume some appropriate oscillating behavior of the nonlinear term either at infinity or at zero. This methodology has been usefully used in obtaining the existence of multiple results for different kinds of problems, such as p-Laplacian problem [12], quasilinear elliptic system [13, 14], discrete BVP [15], double Sturm-Liouville problem [16], and elliptic problems with variable exponent [17]. By using this methodology, to the best of our knowledge, it seems that no similar results are obtained in the literature for fractional BVP. Therefore, the purpose of our paper is to establish the existence of infinitely many solutions for problem (1) by using this type of methodology.Our main results are stated as follows. For this matter, put(3)A=liminfξ→+∞∫0T‍max|x|≤ξF(t,x)dtξ2,B=limsup|x|→+∞∫T/43T/4‍F(t,x)dt|x|2,C(T,α)=∫0T/4‍t2-2αdt+∫T/43T/4‍[t1-α-(t-T4)1-α]2dt+∫3T/4T‍[t1-α-(t-T4)1-α+(t-3T4)1-α]2dt.
Our first main result is the following theorem.Theorem 1.
SupposeF(t,x) satisfies the following conditions.(F0)
For allx∈ℝN, the function t→F(t,x) is measurable.(F1)
For almost allt∈[0,T], the function x→F(t,x) is locally Lipschitz and  F(t,0)=0.(F2)
There exista,b∈L1([0,T],ℝ+) such that |x*|≤a(t)+b(t)|x|r-1 with r∈[1,+∞)  for all x*∈∂F(t,x), all x∈ℝN, and almost all t∈[0,T].(F3)
F(t,x)≥0 for almost all t∈[0,T] and all x∈ℝN.(F4)
A<κB, where κ=(2α-1)T2Γ2(2-α)Γ2(α)/32T2α-1C(T,α).
Then, for eachλ∈(16C(T,α)/BT2Γ2(2-α),(2α-1)Γ2(α)/2AT2α-1), problem (1) admits a sequence of solutions which is unbounded in X.Next, we present the other main result. First, put(4)A1=liminfξ→0+∫0T‍max|x|≤ξF(t,x)dtξ2,B1=limsup|x|→0+∫T/43T/4‍F(t,x)dt|x|2.Theorem 2.
SupposeF(t,x) satisfies the conditions (F0)–(F3), and(F5)
A1<κB1, where κ=(2α-1)T2Γ2(2-α)Γ2(α)/32T2α-1C(T,α).
Then, for eachλ∈(16C(T,α)/B1T2Γ2(2-α),(2α-1)Γ2(α)/2A1T2α-1), problem (1) admits a sequence of pairwise distinct solutions which strongly converges to zero in X.In order to prove Theorems1 and 2, we recall the critical point theorem in [9] here for the readers’ convenience.Theorem 3.
LetX be a reflexive real Banach space, and let Φ,Ψ:X→ℝ be two Lipschitz functions such that Φ is sequentially weakly lower semicontinuous and coercive and Ψ is sequentially weakly upper semicontinuous. For every r>infXΦ, one puts
(5)φ(r)=infu∈Φ-1((-∞,r))supv∈Φ-1((-∞,r))Ψ(v)-Ψ(u)r-Φ(u),γ=liminfr→+∞φ(r),δ=liminfr→(infXΦ)+φ(r).
Then,(a)
ifγ<+∞, for each λ∈(0,1/γ), the following alternative holds: either
(i)
Iλ=Φ-λΨ possesses a global minimum, or(ii)
there is a sequence{un} of critical points (local minimum) of Iλ such that limn→∞Φ(un)=+∞.(b)
Ifδ<+∞, for each λ∈(0,1/δ), the following alternative holds: either
(i)
there is a global minimum ofΦ which is a local minimum of Iλ, or(ii)
there is a sequence{un} of pairwise distinct critical points (local minimum) of Iλ with limn→∞Φ(un)=infXΦ, which weakly converges to a global minimum of Φ.The present paper is organized as follows. In Section2 we present some basic definitions and facts from the nonsmooth analysis theory, and we prove a variational principle for problem (1). Section 3 is devoted to proving Theorems 1 and 2.


## 2. Preliminaries

### 2.1. Nonsmooth Analysis

LetX be a real Banach space and X* its conjugate space; we denote by ∥·∥ and 〈·,·〉, respectively, the norm and the duality pairing between X* and X.For a locally Lipschitz functionφ:X→ℝ, we define the generalized directional derivative of φ at point u in the direction h∈E as follows:
(6)φ0(u;h)=limsupv→0,s↓0φ(u+v+sh)-φ(u+v)s.The generalized gradient of a locally Lipschitz functionφ at the point u, denoted by ∂φ(u), is the set ∂φ(u)={w∈E*:〈w,v〉≤φ0(u;v),∀v∈X}. If φ∈C1(X), then ∂φ(u)={φ′(u)} for all u∈X.A pointu∈X is said to be a critical point of a locally Lipschitz function φ:X→ℝ if 0∈∂φ(u). Clearly, if u is a minimum of a locally Lipschitz function φ, then 0∈∂φ(u); that is, u is a critical point of φ.


### 2.2. Fractional Derivative Space

Throughout this paper, we denote the norm of the spaceLp([0,T],ℝN) for 1≤p≤+∞ as ∥u∥Lp=(∫0T‍|u(t)|pdt)1/p and ∥u∥∞=maxu∈[0,T]|u(t)|.Definition 4.
Let0<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
(7)∥u∥α,p=(∫0T‍|u(t)|pdt+∫0T‍|0cDtαu(t)|pdt)1/p,hhhhhhhhhhhhhhhhhhhhh∀u∈E0α,p.Remark 5.
It is obvious that this fractional derivative spaceE0α,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.The properties of the fractional derivative spacesE0α,p are listed as the following lemma.Lemma 6 (see [8]).
Let0<α≤1 and 1<p<∞.(1)
The fractional derivative spacesE0α,p are a reflexive and separable Banach space.(2)
Ifα>1/p, for any u∈E0α,p, one has 0Dt-α(0Dtαu(t))=u(t), for any t∈[0,T].(3)
If1-α≥1/p or α>1/p, one has E0α,p↪Lp([0,T],ℝN) is compact and
(8)∥u∥Lp≤TαΓ(α+1)∥0cDtαu∥Lp.(4)
Assume thatα>1/p and the sequence {un} converges weakly to u in E0α,p; that is, un⇀u. Then {un} converges strongly to u in C([0,T],ℝN); that is, ∥un-u∥∞→0, as n→+∞. Moreover, if 1/p+1/q=1, one has
(9)∥u∥∞≤Tα-1/pΓ(α)((α-1)q+1)1/q∥0cDtαu∥Lp.According to (8), we can consider E0α,p with respect to the following norm:
(10)∥u∥α,p=∥0cDtαu∥Lp=(∫0T‍|0cDtαu(t)|pdt)1/p.
In this paper, the work space for problem (1) is E0α,2:=Eα. The space Eα is a Hilbert space with respect to the norm ∥u∥=∥u∥α,2 given by (10), and the corresponding inner product is defined by the following:
(11)〈u,v〉=∫0T‍(0cDtαu(t),0cDtαv(t))dt.


### 2.3. Variational Framework

We first give the definition for the solution of problem (1).Definition 7.
A functionu:[0,T]→ℝN is called a solution of problem (1) if tDTα-1(0Dtαu(t)) and 0Dtα-1u(t) are derivatives for almost all t∈[0,T] and u satisfies (1).The functionalIλ:Eα→ℝ corresponding to problem (1) is defined by the following:
(12)Iλ(u)=12∫0T‍|o0Dtαu(t)|2dt-λ∫0T‍F(t,u(t))dt.
By the conditions (F0)–(F2), it is easy to check that Iλ is locally Lipschitz on Eα. Moreover, we can get the variational principle as follows.Proposition 8.
Every critical pointu∈Eα of Iλ is a solution of problem (1).Proof.
We assume thatu∈Eα is a critical point of Iλ; that is 0∈∂Iλ(u); then,
(13)∫0T‍(o0Dtαu(t),0Dtαv(t))dt-λ∫0T‍(w(t),v(t))dt=0,HHHHHHHHHHHHHHHHHHHHHHHH∀v∈Eα
with some w∈∂F(t,u). Noting that w∈L1([0,T],ℝN), then tDT-αw∈L1([0,T],ℝN). Let z(t)=tDT-αw(t), t∈[0,T]. Hence, by the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we have the following:
(14)∫0T‍(z(t),0Dtαv(t))dt=∫0T‍(otDTαz(t),v(t))dt=∫0T‍(otDTα(otDT-αw(t)),v(t))dt=∫0T‍(w(t),v(t))dt.
By (13), for every v∈Eα and hence for every v∈C0∞([0,T],ℝN), we have
(15)∫0T‍(o0Dtαu(t)-λz(t),0Dtαv(t))dt=0.
Sincev∈C0∞([0,T],ℝN), we have 0Dtαv(t)=0Dtα-1v′(t). By the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we get
(16)∫0T‍(otDTα-1(o0Dtαu(t)-λz(t)),v′(t))dt=0,hhhhhhhhhhhhhhh.l∀v∈C0∞([0,T],ℝN).
Since tDTα-1(0Dtαu-λz)∈L1([0,T],ℝN), the standard Fourier series theory implies that
(17)tDTα-1(o0Dtαu(t)-λz(t))=C0,foralmostallt∈[0,T]
for some constant C0∈ℝN. Using the properties of the left and right Riemann-Liouville fractional derivatives, we have
(18)0Dtαu(t)=λz(t)+C(T-t)α-1,a.e.t∈[0,T].
Since w(t)∈L1([0,T],ℝN), we can identify the equivalence class tDTα-1(0Dtαu) and its continuous representant
(19)tDTα-1(o0Dtαu(t))=λ∫tT‍w(t)dt+C,fort∈[0,T].
Firstly, we notice that 0Dtα-1u(t) is derivative for almost every t∈[0,T] and (0Dtα-1u(t))′=0Dtαu(t)∈L2([0,T],ℝN) as u∈Eα. On the other hand, tDTα-1(0Dtαu(t)) is derivative a.e. on [0,T] and (tDTα-1(0Dtαu(t)))′∈L1([0,T],ℝN).
By (19), we have
(20)tDTα(o0Dtαu(t))=-(otDTα-1(o0Dtαu(t)))′=λw(t),hhhhhhhhhhlhh.a.e.t∈[0,T].
Moreover, u∈Eα implies that u(0)=u(T)=0. The proof is completed.


## 2.1. Nonsmooth Analysis

LetX be a real Banach space and X* its conjugate space; we denote by ∥·∥ and 〈·,·〉, respectively, the norm and the duality pairing between X* and X.For a locally Lipschitz functionφ:X→ℝ, we define the generalized directional derivative of φ at point u in the direction h∈E as follows:
(6)φ0(u;h)=limsupv→0,s↓0φ(u+v+sh)-φ(u+v)s.The generalized gradient of a locally Lipschitz functionφ at the point u, denoted by ∂φ(u), is the set ∂φ(u)={w∈E*:〈w,v〉≤φ0(u;v),∀v∈X}. If φ∈C1(X), then ∂φ(u)={φ′(u)} for all u∈X.A pointu∈X is said to be a critical point of a locally Lipschitz function φ:X→ℝ if 0∈∂φ(u). Clearly, if u is a minimum of a locally Lipschitz function φ, then 0∈∂φ(u); that is, u is a critical point of φ.


## 2.2. Fractional Derivative Space

Throughout this paper, we denote the norm of the spaceLp([0,T],ℝN) for 1≤p≤+∞ as ∥u∥Lp=(∫0T‍|u(t)|pdt)1/p and ∥u∥∞=maxu∈[0,T]|u(t)|.Definition 4.
Let0<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
(7)∥u∥α,p=(∫0T‍|u(t)|pdt+∫0T‍|0cDtαu(t)|pdt)1/p,hhhhhhhhhhhhhhhhhhhhh∀u∈E0α,p.Remark 5.
It is obvious that this fractional derivative spaceE0α,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.The properties of the fractional derivative spacesE0α,p are listed as the following lemma.Lemma 6 (see [8]).
Let0<α≤1 and 1<p<∞.(1)
The fractional derivative spacesE0α,p are a reflexive and separable Banach space.(2)
Ifα>1/p, for any u∈E0α,p, one has 0Dt-α(0Dtαu(t))=u(t), for any t∈[0,T].(3)
If1-α≥1/p or α>1/p, one has E0α,p↪Lp([0,T],ℝN) is compact and
(8)∥u∥Lp≤TαΓ(α+1)∥0cDtαu∥Lp.(4)
Assume thatα>1/p and the sequence {un} converges weakly to u in E0α,p; that is, un⇀u. Then {un} converges strongly to u in C([0,T],ℝN); that is, ∥un-u∥∞→0, as n→+∞. Moreover, if 1/p+1/q=1, one has
(9)∥u∥∞≤Tα-1/pΓ(α)((α-1)q+1)1/q∥0cDtαu∥Lp.According to (8), we can consider E0α,p with respect to the following norm:
(10)∥u∥α,p=∥0cDtαu∥Lp=(∫0T‍|0cDtαu(t)|pdt)1/p.
In this paper, the work space for problem (1) is E0α,2:=Eα. The space Eα is a Hilbert space with respect to the norm ∥u∥=∥u∥α,2 given by (10), and the corresponding inner product is defined by the following:
(11)〈u,v〉=∫0T‍(0cDtαu(t),0cDtαv(t))dt.


## 2.3. Variational Framework

We first give the definition for the solution of problem (1).Definition 7.
A functionu:[0,T]→ℝN is called a solution of problem (1) if tDTα-1(0Dtαu(t)) and 0Dtα-1u(t) are derivatives for almost all t∈[0,T] and u satisfies (1).The functionalIλ:Eα→ℝ corresponding to problem (1) is defined by the following:
(12)Iλ(u)=12∫0T‍|o0Dtαu(t)|2dt-λ∫0T‍F(t,u(t))dt.
By the conditions (F0)–(F2), it is easy to check that Iλ is locally Lipschitz on Eα. Moreover, we can get the variational principle as follows.Proposition 8.
Every critical pointu∈Eα of Iλ is a solution of problem (1).Proof.
We assume thatu∈Eα is a critical point of Iλ; that is 0∈∂Iλ(u); then,
(13)∫0T‍(o0Dtαu(t),0Dtαv(t))dt-λ∫0T‍(w(t),v(t))dt=0,HHHHHHHHHHHHHHHHHHHHHHHH∀v∈Eα
with some w∈∂F(t,u). Noting that w∈L1([0,T],ℝN), then tDT-αw∈L1([0,T],ℝN). Let z(t)=tDT-αw(t), t∈[0,T]. Hence, by the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we have the following:
(14)∫0T‍(z(t),0Dtαv(t))dt=∫0T‍(otDTαz(t),v(t))dt=∫0T‍(otDTα(otDT-αw(t)),v(t))dt=∫0T‍(w(t),v(t))dt.
By (13), for every v∈Eα and hence for every v∈C0∞([0,T],ℝN), we have
(15)∫0T‍(o0Dtαu(t)-λz(t),0Dtαv(t))dt=0.
Sincev∈C0∞([0,T],ℝN), we have 0Dtαv(t)=0Dtα-1v′(t). By the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we get
(16)∫0T‍(otDTα-1(o0Dtαu(t)-λz(t)),v′(t))dt=0,hhhhhhhhhhhhhhh.l∀v∈C0∞([0,T],ℝN).
Since tDTα-1(0Dtαu-λz)∈L1([0,T],ℝN), the standard Fourier series theory implies that
(17)tDTα-1(o0Dtαu(t)-λz(t))=C0,foralmostallt∈[0,T]
for some constant C0∈ℝN. Using the properties of the left and right Riemann-Liouville fractional derivatives, we have
(18)0Dtαu(t)=λz(t)+C(T-t)α-1,a.e.t∈[0,T].
Since w(t)∈L1([0,T],ℝN), we can identify the equivalence class tDTα-1(0Dtαu) and its continuous representant
(19)tDTα-1(o0Dtαu(t))=λ∫tT‍w(t)dt+C,fort∈[0,T].
Firstly, we notice that 0Dtα-1u(t) is derivative for almost every t∈[0,T] and (0Dtα-1u(t))′=0Dtαu(t)∈L2([0,T],ℝN) as u∈Eα. On the other hand, tDTα-1(0Dtαu(t)) is derivative a.e. on [0,T] and (tDTα-1(0Dtαu(t)))′∈L1([0,T],ℝN).
By (19), we have
(20)tDTα(o0Dtαu(t))=-(otDTα-1(o0Dtαu(t)))′=λw(t),hhhhhhhhhhlhh.a.e.t∈[0,T].
Moreover, u∈Eα implies that u(0)=u(T)=0. The proof is completed.


## 3. Proof of Main Results

Throughout this section, foru∈X:=Eα, we denote Iλ(u)=Φ(u)-λΨ(u), where λ∈(16C(T,α)/BT2Γ2(2-α),(2α-1)Γ2(α)/2AT2α-1), and
(21)Φ(u)=12∥u∥2,Ψ(u)=∫0T‍F(t,u(t))dt.
Clearly, Φ is Gâteaux differentiable and sequentially weakly lower semicontinuous and coercive; Ψ is locally Lipschitz continuous on X; by standard argument, Ψ is sequentially weakly continuous. We denote
(22)A≔liminfξ→+∞∫0T‍max|x|≤ξF(t,x)dtξ2,B:=limsup|x|→+∞∫0T‍F(t,x)dt|x|2.Proof of Theorem1.
First, we verify thatγ<+∞. By (F4), let {an} be a real sequence such that limn→∞an=+∞ and
(23)limn→∞∫0T‍max|x|≤anF(t,x)dtan2=A.
Put rn=((2α-1)Γ2(α)/2T2α-1)an2 for all n∈ℕ. From (9), one has ∥u∥∞≤an for all u∈X such that ∥u∥2≤2rn. Take into account that ∥u0∥=0 and ∫0T‍F(t,0)dt=0, where u0(t)=0 for all t∈[0,T]. For all n∈ℕ, we have
(24)φ(rn)=inf∥u∥2<2rnsup∥v∥2≤2rn∫0T‍F(t,v(t))dt-∫0T‍F(t,u(t))dtrn-∥u∥2/2≤sup∥v∥2≤2rn∫0T‍F(t,v(t))dtrn≤2T2α-1(2α-1)Γ2(α)∫0T‍max|x|≤anF(t,x)dtan2.
Since from assumption (F4) one has A<+∞, then we have
(25)γ≤liminfn→+∞φ(rn)≤2T2α-1(2α-1)Γ2(α)A<+∞.
Now fixλ∈(16C(T,α)/BT2Γ2(2-α),(2α-1)Γ2(α)/2AT2α-1). We claim that the functional Iλ is unbounded from below.
By(F4), let {bn} be a sequence of ℝN such that |bn|→+∞ and
(26)limn→+∞∫T/43T/4‍F(t,bn)dt|bn|2=B.
For eachn∈ℕ, we define a sequence {wn} as follows:
(27)wn(t)={4bnTt,t∈[0,T4),bn,t∈[T4,3T4],4bnT(T-t),t∈(3T4,T].
It is easy to check that wn(0)=wn(T)=0 and wn∈L2([0,T]). Moreover, wn(t) is Lipschitz continuous on [0,T], and hence wn(t) is absolutely continuous on [0,T]. By calculations, we get
(28)0Dtαwn(t)={4bnTΓ(2-α)t1-α,t∈[0,T4),4bnTΓ(2-α)[t1-α-(t-T4)1-α],4bnTΓ(2-α)t1-α,.t∈[T4,3T4],4bnTΓ(2-α)[t1-α-(t-T4)1-αhhhhhhhhhh+(t-3T4)1-α],4bnTΓ(2-α)t1-α,t∈(3T4,T].
Obviously, 0Dtαwn(t) is continuous on [0,T] and
(29)∫0T‍|o0Dtαwn(t)|2=16|bn|2T2Γ2(2-α)×{∫0T/4‍t2-2αdthhhhh+∫T/43T/4‍[t1-α-(t-T4)1-α]2dthhhhh+∫3T/4T‍[t1-α-(t-T4)1-αhhhhhhhhhhhh+(t-3T4)1-α]2dt∫0T/4‍t2-2αdt}:=16|bn|2T2Γ2(2-α)C(T,α),
where C(T,α) depends on T and α.
By condition(F3), we have
(30)∫0T‍F(t,wn(t))dt≥∫T/43T/4‍F(t,bn)dt
for all n∈ℕ. Then,
(31)Iλ(wn)=Φ(wn)-λΨ(wn)≤16|bn|2T2Γ2(2-α)C(T,α)-λ∫T/43T/4‍F(t,bn)dt
for every n∈ℕ.
IfB<+∞, let ε∈(16C(T,α)/λBT2Γ2(2-α),1); by (26) there exists Nε>0 such that
(32)∫T/43T/4‍F(t,bn)dt≥εB|bn|2
for all n≥Nε. Hence, by (31) and (32), we obtain
(33)Iλ(wn)≤[16T2Γ2(2-α)C(T,α)-λεB]|bn|2
for all n≥Nε. Choosing suitable ε, we have
(34)limn→+∞Iλ(wn)=-∞.
On the other hand, if B=+∞, we fix M>16C(T,α)/λT2Γ2(2-α), and again from (26) there exists NM∈ℕ such that
(35)∫T/43T/4‍F(t,bn)dt≥M|bn|2
for all n>NM. Therefore, from (31) and (35), we have
(36)Iλ(wn)≤[16T2Γ2(2-α)C(T,α)-λM]|bn|2
for all n>NM. From the choice of M, we have
(37)limn→+∞Iλ(wn)=-∞.
Hence, our claim is proved.
Since all the assumptions of the case  (a) of Theorem3 are verified, for each λ∈(16C(T,α)/BT2Γ2(2-α),(2α-1)Γ2(α)/2AT2α-1), the functional Iλ admits an unbounded sequence of critical points. The conclusion follows from Proposition 8.Proof of Theorem2.
The proof is the same as Theorem1 by using the case   (b) of Theorem 3 instead of the case   (a).Example 9.
We give an example to illustrate Theorem1.
Set(38)an=n!n,bn=n!(n+12)
for every n∈ℕ. Define the nonnegative function F(t,x):[0,T]×ℝN→ℝ as follows:
(39)F(t,x)={(|x|-n!2)2e-|1/((|x|-n!n-n!/2)2-(n!/2)2)|cosπ2Tt,hhhhhhhhhhhhhhhhlifn!n<|x|<(n+1)!,0,hhhhhhhhhhhhhhhotherwise.
Obviously, F(t,x) satisfies the conditions (F0)–(F3). Next, we show that (F4) is true. Indeed, by direct computation, we get
(40)limn→∞∫0T‍max|x|≤anF(t,x)dtan2=0,limn→∞∫T/43T/4‍F(t,bn)dtbn2=π2T(sin3π8-sinπ8).
Hence, we see that
(41)A=liminfξ→+∞∫0T‍max|x|≤ξF(t,x)dtξ2=0,0<B=limsup|x|→+∞∫T/43T/4‍F(t,x)dt|x|2<+∞.
Therefore, (F4) is verified.



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*Source: 181052-2013-09-16.xml*