Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 468980, 16 pages
doi:10.1155/2012/468980
Research Article
Multiple Solutions for a Class of Fractional
Boundary Value Problems
Ge Bin
Department of Mathematics, Harbin Engineering University, Harbin 150001, China
Correspondence should be addressed to Ge Bin, gebin04523080261@163.com
Received 6 March 2012; Accepted 19 September 2012
Academic Editor: Yong H. Wu
Copyright q 2012 Ge Bin. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We study the multiplicity of solutions for the following fractional boundary value problem:
−β
−β
d/dt1/2 0 Dt u t 1/2 0 DT u t λ∇Ft, ut 0, a.e. t ∈ 0, T , u0 uT 0,
−β
−β
where 0 Dt and 0 DT are the left and right Riemann-Liouville fractional integrals of order
0 ≤ β < 1, respectively, λ > 0 is a real number, F : 0, T × RN → R is a given function, and
∇Ft, x is the gradient of F at x. The approach used in this paper is the variational method. More
precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of
at least two nontrivial solutions.
1. Introduction
In this paper, we consider the fractional boundary value problem of the following form:
d 1 −β 1 −β u t 0 DT u t λ∇Ft, ut 0,
0D
dt 2 t
2
a.a. t ∈ 0, T ,
P u0 uT 0,
−β
−β
where 0 Dt and 0 DT are the left and right Riemann-Liouville fractional integrals of order
0 ≤ β < 1, respectively, λ > 0 is a real number, F : 0, T × RN → R is a given function, and
∇Ft, x is the gradient of F at x.
2
Abstract and Applied Analysis
In particular, if λ 1, the problem P1 reduces to the standard second-order boundary
value problem of the following form:
d 1 −β 1 −β D
D
∇Ft, ut 0, a.a. t ∈ 0, T ,
u
u
t
t
0
0
dt 2 t
2 T
P1 u0 uT 0.
Fractional calculus and fractional differential equations can find many applications
in various fields of physical science such as viscoelasticity, diffusion, control, relaxation
processes, and modeling phenomena in engineering, see 1–12. Recently, many results were
obtained dealing with the existence and multiplicity of solutions of nonlinear fractional
differential equations by use of techniques of nonlinear analysis, such as fixed-point theory
including Leray-Schauder nonlinear alternative see 13, 14, topological degree theory
including coincidence degree theory see 15, 16, and comparison method including
upper and lower solutions methods and monotone iterative method see 17, 18. However,
it seems that the popular methods mentioned above are not appropriate for discussing P and P1 , as the equivalent integral equation is not easy to be obtained.
In the past, there were investigations of the eigenvalue problems for fractional
differential equations. For more detailed information on this topic, we refer to Zhang et al.
19–21, Wang et al. 22, 23, and Jiang et al. 24.
Recently, there are many papers dealing with the existence of solutions for problem
P1 . In 25, Jiao and Zhou obtained the existence of solutions for P1 by mountain
pass theorem under the Ambrosetti-Rabinowitz condition. Chen and Tang 26 studied the
existence and multiplicity of solutions for the system P1 when the nonlinearity Ft, · is
superquadratic, asymptotically quadratic, and subquadratic, respectively.
But so far, few papers discuss the two solutions of the system P via critical point
theory. The aim of the present paper is to study the existence of at least two solutions for the
system P as the parameter λ > λ0 for some constant λ0 .
The paper is organized as follows. We first introduce some basic preliminary results
and a well-known lemma in Section 2, including the fractional derivative space Eα , where
α ∈ 1/2, 1. In Section 3, we give the main result and its proof. In Section 4, we give the
summary of this paper.
2. Preliminary
In this section, we recall some related preliminaries and display the variational setting which
has been established for our problem.
Definition 2.1 see 8. Let ft be a function defined on a, b and τ > 0. The left and right
Riemann-Liouville fractional integrals of order τ for function ft denoted by a D−τ
t ft and
−τ
D
ft,
respectively,
are
defined
by
t b
t
1
−τ
t − sτ−1 fsds, t ∈ a, b,
a D t ft Γτ a
2.1
b
1
τ−1
−τ
D
ft
fsds,
t
∈
b,
−
s
a,
t
t b
Γτ t
provided the right-hand sides are pointwise defined on a, b, where Γ is the gamma function.
Abstract and Applied Analysis
3
Definition 2.2 see 8. Let ft be a function defined on a, b. The left and right RiemannLiouville fractional derivatives of order τ for function ft denoted by a Dτb ft and t Dτb ft,
respectively, are defined by
τ
a Dt ft
τ
t Db ft
dn
dn
1
n a Dτ−n
t ft dt
Γn − τ dtn
t
t − s
fsds ,
a
dn
dn τ−n
1
−1
t D b ft n
dt
Γn − τ dtn
n
n−τ−1
b
n−τ−1
t − s
2.2
fsds ,
t
where t ∈ a, b, n − 1 ≤ τ < n, and n ∈ N.
The left and right Caputo fractional derivatives are defined via the above RiemannLiouville fractional derivatives. In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by ACa, b, RN .
ACk a, b, RN k 1, 2, . . . is the space of functions f such that f ∈ Ck a, b, RN . In
particular, ACa, b, RN AC1 a, b, RN .
Definition 2.3 see 8. Let τ ≥ 0 and n ∈ N. If τ ∈ n − 1, n and ft ∈ ACn a, b, RN ,
then the left and right Caputo fractional derivatives of order τ for function ft denoted
by ca Dtτ ft and ct Dbτ ft, respectively, which exist a.e. on a, b. ca Dtτ ft and ct Dbτ ft are
represented by
c τ
a D t ft
c τ
t D b ft
τ−n n
t
a Dt f
1
Γn − τ
n
−1 nt Dτ−n
t
b f
t
1
Γn − τ
t − s
n−τ−1 n
f
sds ,
a
b
t − s
n−τ−1 n
f
2.3
sds ,
t
respectively, where t ∈ a, b.
α,p
Definition 2.4 see 25. Define 0 < α ≤ 1 and 1 < p < ∞. The fractional derivative space E0
is defined by the closure of C0∞ 0, T , RN with respect to the norm
uα,p T
0
p
|ut| dt T
0
1/p
c α
D utp dt
0
t
,
α,p
∀u ∈ E0 ,
2.4
where C0∞ 0, T , RN denotes the set of all functions u ∈ C∞ 0, T , RN with u0 α,p
uT 0. It is obvious that the fractional derivative space E0 is the space of functions
p
N
u ∈ L 0, T , R having an α-order Caputo fractional derivative c0 Dαt u ∈ Lp 0, T , RN and u0 uT 0.
α,p
Proposition 2.5 see 25. Let 0 < α ≤ 1 and 1 < p < ∞. The fractional derivative space E0 is a
reflexive and separable space.
4
Abstract and Applied Analysis
α,p
Proposition 2.6 see 25. Let 0 < α ≤ 1 and 1 < p < ∞. For all u ∈ E0 , one has
uLp ≤
Tα c Dα u
p .
0 t
L
Γα 1
2.5
Moreover, if α > 1/p and 1/p 1/q 1, then
u∞ ≤
c α 1/q 0 Dt u
Lp .
Γα α − 1q 1
T α−1/p
2.6
α,p
According to 2.6, one can consider E0 with respect to the norm
uα,p
c Dα u
0
t
Lp
T
0
1/p
c α p
D u dt
0
t
2.7
.
Proposition 2.7 see 25. Define 0 < α ≤ 1 and 1 < p < ∞. Assume that α > 1/p, and the
α,p
sequence uk converges weakly to u ∈ E0 , that is, uk u. Then uk → u in C0, T , RN , that is,
uk − u∞ → 0, as k → ∞.
Making use of Definition 2.3, for any u ∈ AC0, T , RN , problem P is equivalent to
the following problem:
1 α−1 c α
d 1
α−1 c α
λ∇Ft, ut 0,
0 Dt
t D T ut
0 D t ut − t D T
dt 2
2
a.e. t ∈ 0, T ,
P2 u0 uT 0,
where α 1 − β/2 ∈ 1/2, 1.
In the following, we will treat problem P2 in the Hilbert space Eα E0α,2 with the
corresponding norm uα uα,2 . It follows from 25, Theorem 4.1 that the functional ϕ
given by
ϕu T
0
T
1 c α
c α
Ft, utdt
− 0 Dt ut, t DT ut dt − λ
2
0
2.8
is continuously differentiable on Eα . Moreover, for u, v ∈ Eα , we have
ϕ u, v −
T
0
−λ
c α
1 c α
c α
c α
0 D t ut, t D T vt t D T ut, 0 D t vt dt
2
T
2.9
∇Ft, ut, vtdt.
0
Definition 2.8 see 25. A function u ∈ AC0, T , RN is called a solution of P2 if
i Dα ut is derivative for a.e. t ∈ 0, T ,
c α
α−1 c α
ii u satisfies P2 , where Dα ut : 1/2 0 Dα−1
t 0 Dt ut − 1/2 t DT t DT ut.
Abstract and Applied Analysis
5
Proposition 2.9 see 25. If 1/2 < α ≤ 1, then for any u ∈ Eα , one has
|cosπα|u2α
≤−
T
0
α
c α
0 D t ut, t DT ut dt
c
≤
1
u2α .
|cosπα|
2.10
Proposition 2.10 see 25. Let 1/2 < α ≤ 1 be satisfied. If u ∈ Eα , then the functional J : Eα → R
denoted by
1
Ju −
2
T
0
α
c α
0 D t ut, t D T ut dt
c
2.11
is convex and continuous on Eα .
In order to prove the existence of two solutions for problem P , firstly, we recall some
well-known results. Their proofs can be found in many books. Please refer to the references
and its references therein.
Lemma 2.11 see 27. If X is a Banach space, ϕ ∈ C1 X, R, e ∈ X, and r > 0, such that e > r
and
b : inf ϕu > ϕ0 ≥ ϕe,
ur
2.12
and if ϕ satisfies the PS condition, with
c : inf max ϕ γt ,
γ∈Γ t∈0,1
Γ : γ ∈ C0, 1, X : γ0 0, γ1 e ,
2.13
then c is a critical value of ϕ.
3. The Main Result and Proof of the Theorem
In this part, we will prove that for P there also exist two solutions for the general case.
Our hypotheses on nonsmooth potential Ft, x are as follows.
HF1 : F : 0, T × RN → R is a function such that Ft, 0 0 a.e. on 0, T and satisfies
the following facts:
i for all x ∈ RN , t → Ft, x is measurable,
ii for a.a. t ∈ 0, T , x → Ft, x is continuously differentiable,
iii there exist c ∈ C0, T , R and 0 < α0 < 2, such that
|∇Ft, x|, |Ft, x| ≤ ct 1 |x|α0 ,
for a.a. t ∈ 0, T , all x ∈ RN ,
3.1
iv there exist γ > 2 and μ ∈ L∞ 0, T , such that
lim sup
|x| → 0
∇Ft, x, x
< μt,
|x|γ
uniformly for a.a. t ∈ 0, T ,
3.2
6
Abstract and Applied Analysis
v there exist ξ0 ∈ RN , t0 ∈ 0, T , and r0 > 0, such that
Ft, ξ0 > δ0 > 0,
a.e. t ∈ Br0 t0 ,
3.3
where Br0 t0 : {t ∈ 0, T : |t − t0 | ≤ r0 } ⊂ 0, T .
Remark 3.1. It is easy to verify that F : 0, T × RN → R satisfies the whole assumption in 25,
Theorem 4.1. So, u ∈ Eα is a solution of the corresponding Euler equation ϕ u 0, then u is
a solution of problem P2 which, of course, corresponds to the solution of problem P . For
great details, please see 25, Theorem 4.2.
Theorem 3.2. Suppose that HF1 holds. Then there exists λ0 > 0 such that for each λ > λ0 , the problem P2 has at least two nontrivial solutions, which correspond to the two solutions of problem P .
Proof. The proof is divided into four steps as follows.
Step 1. We will show that ϕ is coercive in this step.
Firstly, by HF1 iii, 2.6, and 2.10, we have
T
T
1 c α
c α
ϕu Ft, utdt
− 0 Dt ut, t DT ut dt − λ
2
0
0
T
|cosπα|
≥
|ut|α0 dt
u2α − c0 λT − c0 λ
2
0
α0
T α−1/2
|cosπα|
2
≥
uαα0
uα − c0 λT − c0 λT √
2
2α − 1Γα
−→ ∞,
3.4
as uα −→ ∞,
where c0 maxt∈0,T |ct|.
Step 2. We will show that the ϕ is weakly lower semicontinuous.
Let un u weakly in Eα , and by Proposition 2.7, we obtain the following results:
Eα → C 0, T , RN ,
un t −→ ut
for a.e. t ∈ 0, T ,
Ft, un t −→ Ft, ut
3.5
for a.e. t ∈ 0, T .
By Fatou’s lemma,
T
lim sup
n→∞
0
Ft, un tdt ≤
T
Ft, utdt.
3.6
0
On the other hand, by Proposition 2.10, we have limn → ∞ Jun Ju, that is,
T
lim
n→∞
0
T
1 c α
1 c α
c α
c α
− 0 Dt un t, t DT un t dt − 0 Dt ut, t DT ut dt.
2
2
0
3.7
Abstract and Applied Analysis
7
Thus,
T
T
1 c α
c α
lim inf ϕun lim inf
Ft, un tdt
− 0 Dt un t, t DT un t dt − lim sup λ
n→∞
n→∞
2
n→∞
0
0
T
T
1
≥
− c0 Dαt ut, ct DαT ut dt − λ
Ft, utdt ϕu.
2
0
0
3.8
Hence, by the Weierstrass theorem, we deduce that there exists a global minimizer
u0 ∈ Eα such that
ϕu0 minα ϕu.
3.9
u∈E
Step 3. We will show that there exists λ0 > 0 such that for each λ > λ0 , ϕu0 < 0.
By the condition by HF1 v, there exists ξ0 ∈ RN such that Ft, ξ0 > δ0 > 0, a.e.
t ∈ Br0 t0 . It is clear that
0 < M1 : max c0 c0 |x|α0 < ∞.
3.10
|x|≤|ξ |
0
Now we denote
κ0 M1
,
δ0 M1
3.11
|ξ0 |2 T 3−2α
λ0 max
κ∈κ1 ,κ2 2Γ2 1
− αr03 1 − κ2 1 − α2 3 − 2αδ0 κ − M1 κM1 |cosπα|
,
where κ0 < κ1 < κ2 < 1, and δ0 is given in the condition HF1 v. A simple calculation shows
that the function κ → δ0 κ − M1 κM1 is positive whenever κ > κ0 and δ0 κ0 − M1 κ0 M1 0.
Thus, λ0 is well defined and λ0 > 0.
We will show that for each λ > λ0 , the problem P2 has two nontrivial solutions. In
order to do this, for t ∈ t1 , t2 , let us define
⎧
⎪
0
⎪
⎪
⎨
ηκ t ξ0
⎪
ξ0
⎪
⎪
⎩
r0 − |t − t0 |
r0 1 − κ
if t ∈ 0, T /Br0 t0 ,
if t ∈ Bκr0 t0 ,
3.12
if t ∈ Br0 t0 /Bκr0 t0 .
Then |ηκ t| ≤ |ξ0 |/r0 1 − κ and
c α
D ηκ t ≤
0
t
1
Γ1 − α
t
0
t − s−α ηκ sds ≤
1
t1−α
|ξ0 |
.
Γ1 − α r0 1 − κ 1 − α
3.13
Hence,
2
ηι α
T
0
c α
D ηκ t2 dt ≤
0
t
|ξ0 |2 T 3−2α
Γ2 1 − αr02 1 − κ2 1 − α2 3 − 2α
.
3.14
8
Abstract and Applied Analysis
By HF1 iii and v, we have
T
F t, ηκ t dt 0
Bκr0 t0 F t, ηκ t dt Br0 t0 \Bκr0 t0 F t, ηκ t dt
≥ 2κr0 δ0 − M1 2r0 − 2κr0 3.15
2r0 δ0 κ − M1 κM1 .
For κ ∈ κ1 , κ2 , by Proposition 2.9, we have
ϕ ηκ T
0
−
T
1 c α
c α
η
η
F t, ηκ t dt
dt
−
λ
t,
t
D
D
t T κ
0 t κ
2
0
≤
2
1
ηκ − 2λr0 δ0 κ − M1 κM1 α
|cosπα|
≤
1
|ξ0 |2 T 3−2α
− 2λr0 δ0 κ − M1 κM1 ,
|cosπα| Γ2 1 − αr 2 1 − κ2 1 − α2 3 − 2α
3.16
0
so that ϕηt < 0 whenever λ > λ0 .
Step 4. We will check the PS condition in the following.
Suppose that {un }n≥1 ⊆ Eα such that ϕun → c and ϕ un α → 0.
Since ϕ is coercive and {un }n≥1 is bounded in Eα and passed to a subsequence, which
still denote {un }n≥1 , we may assume that there exists u ∈ Eα , such that un u weakly in Eα ;
thus, we have
ϕ un − ϕ u, un − u ϕ un , un − u − ϕ u, un − u
≤ ϕ un un α − ϕ u, un − u −→ 0,
3.17
as n → ∞. Moreover, according to Proposition 2.7, we have un − u∞ → 0, as n → ∞.
Observing that
ϕ un − ϕ u, un − u
T
c α
c α
−
0 D t un t − ut, t D T un t − ut dt
0
−
T
∇Ft, un t − Ft, ut, un t − utdt
0
≥ |cosπα|un −
u2α
−
T
0
3.18
|∇Ft, un t − Ft, ut|dtun − u∞ ,
combining this with 3.17, it is easy to verify that un − uα → 0, as n → ∞, and hence that
un → u in Eα . Thus, ϕ satisfies the PS condition.
Abstract and Applied Analysis
9
Step 5. We will show that there exists another nontrivial weak solution of problem P2 .
From the mean value theorem and HF1 v, we have
1
Ft, x − Ft, 0 ∇Ft, θx, xdθ
0
1
0
∇Ft, θx, θx
dθ
θ
1
μt
|θx|γ dθ
0 θ
1
γ
μt|x|
θγ−1 dθ
≤
3.19
0
γ
μt|x|
,
γ
for all |x| < β, a.e. t ∈ 0, T .
It follows from the conditions HF1 iii and all |x| ≥ β and a.e. t ∈ 0, T that
|Ft, x| ≤ c0 |t| c0 |x|α0
x
≤ c0 c0 |x|α0
β
c0
α0 −γ
≤
c
|x|γ
0 |x|
βα0
c0
c0
≤
|x|γ ,
βγ βγ−α0
3.20
and this together with 3.19 yields that for all x ∈ RN and a.e. t ∈ 0, T ,
|Ft, x| ≤
μt c0
c0
γ γ−α0 |x|γx ≤ c1 |x|γ ,
γ
β
β
3.21
for some positive constant c1 .
For all λ > λ0 , uα < 1, and |u|∞ < 1, we have
T
T
1
Ft, utdt
− c0 Dαt ut, ct DTα ut dt − λ
2
0
0
T
≥ |cosπα|u2α − λc1
|ut|γ dt
ϕu ≥
|cosπα|u2α
− λc1
0
T α−1/2
√
Γαα 2α − 1
γ
γ
| uα .
3.22
10
Abstract and Applied Analysis
So, for ρ > 0 small enough, there exists a ν > 0 such that
ϕu > ν,
for uα ρ ,
3.23
and u0 α > ρ. So by the mountain pass theorem cf. Lemma 2.11, we can get u1 ∈ Eα which
satisfies
ϕ u1 0.
ϕu1 c > 0,
3.24
Therefore, u1 is another nontrivial critical point of ϕ.
Remark 3.3. We can find a potential function satisfying the hypothesis of our Theorem 3.2. For
great details, please see Section 4B in Summary.
So far, the results involved potential functions exhibiting sublinear. The next theorem
concerns problems where the potential function is superlinear.
Our hypotheses on nonsmooth potential Ft, x are as follows.
HF2 : F : 0, T × RN → R is a function such that Ft, 0 0 a.e. on 0, T and satisfies
the following facts:
i for all x ∈ RN , t → Ft, x is measurable,
ii for a.a. t ∈ 0, T , x → Ft, x is continuously differentiable,
iii there exist c ∈ C0, T , R and α0 > 2, such that
|∇Ft, x|, |Ft, x| ≤ ct 1 t|x|α0 ,
for a.a. t ∈ 0, T , all x ∈ RN ,
3.25
iv there exist γ > 2 and μ ∈ L∞ 0, T , such that
lim sup
|x| → 0
∇Ft, x, x
< μt,
|x|γ
uniformly for a.a. t ∈ 0, T ,
3.26
v there exist ξ0 ∈ RN , t0 ∈ 0, T , and r0 > 0, such that Ft, ξ0 > δ0 > 0, a.e. t ∈
Br0 t0 ,where Br0 t0 : {t ∈ 0, T : |t − t0 | ≤ r0 } ⊂ 0, T ,
vi for a.a. t ∈ 0, T and all x ∈ RN , we have
Ft, x ≤ νt with ν ∈ Lβ 0, T , R,
1 ≤ β < 2.
3.27
Theorem 3.4. Suppose that HF2 holds. Then there exists a λ0 > 0 such that for each λ > λ0 , the
problem P2 has at least two nontrivial solutions, which correspond to the two solutions of problem
P .
Proof. The steps are similar to those of Theorem 3.2. In fact, we only need to modify Step 1
and Step 4 as follows: 1 shows that ϕ is coercive under the condition HF2 vi; 4 shows that
Abstract and Applied Analysis
11
there exists a second nontrivial solution under the conditions HF2 iii and HF2 iv. Then
from Steps 1 , 2, 3, and 4 above, problem P has at least two nontrivial solutions.
Step 1 . By HF2 vi, for all u ∈ Eα , uα > 1, we have
T
T
1
ϕu Ft, utdt
− c0 Dαt ut, ct DαT ut dt − λ
2
0
0
3.28
T
≥ |cosπα|u2α − λc0
νtdt −→ ∞,
0
as uα −→ ∞,
where c0 maxt∈0,T |ct|.
Step 4 . Because of hypothesis HF2 iii, we have
Ft, x ≤ c0 c0 |x|α0
α0
x
≤ c0 c0 |x|α0
β
α0
1
c0 |x|α0 c0 |x|α0
β
3.29
c2 |x|α0 ,
for a.e. t ∈ 0, T and all |x| ≥ β with c2 > 0.
Combining 3.19 and 3.29, it follows that
|Ft, x| ≤
μt γ
|x| c2 |x|α0 ,
γ
for a.e. t ∈ 0, T and all x ∈ RN .
Thus, for all λ > λ0 , uα < 1 and |u|∞ < 1, we have
T
T
1 c α
c α
Ft, utdt
− 0 Dt ut, t DT ut dt − λ
ϕu 2
0
0
T
T
μt
γ
2
≥ |cosπα|uα − λ
|ut|α0 dt
|ut| dt − λc2
γ
0
0
γ α0 T α−1/2
T α−1/2
γ
2
≥ |cosπα|uα − λc3
√
√
uα − λc4
uαα0 ,
Γα 2α − 1
Γα 2α − 1 3.30
3.31
where c3 and c4 are positive constants.
So, for ρ > 0 small enough, there exists a ν > 0 such that
ϕu > ν,
for uα ρ ,
3.32
and u0 α > ρ.
Arguing as in proof of Step 4 of Theorem 3.2, we conclude that ϕ satisfies the PS
condition. So by the mountain pass theorem cf. Lemma 2.11, we can get that u1 ∈ Eα satisfies
ϕu1 c > 0,
ϕ u1 0.
3.33
12
Abstract and Applied Analysis
Therefore, u1 is another nontrivial critical point of ϕ.
Remark 3.5. We will give some examples, which satisfy the hypothesis of our Theorem 3.4.
For great details, please see Section 4C in Summary.
4. Summary
A If β 0, then α 1 − β/2 1. Therefore, by Theorems 3.2 and 3.4, we actually obtain the
existence of two weak solutions of the following eigenvalue problem:
u t λ∇Ft, ut 0,
a.a. t ∈ 0, T ,
P3 u0 uT 0,
where λ > 0 is a real number, F : 0, T × RN → R is a given function, and ∇Ft, x is the
gradient of F at x. Although many excellent results have been worked out on the existence
of solutions for second-order system P3 e.g., 28, 29, it seems that no similar results were
obtained in the literature for fractional system P .
B We give an example in the following to illustrate our viewpoint in Remark 3.1. We
consider
⎧
π
⎪
|x|γ ,
⎨at sin
2
Ft, x ⎪
⎩at,
0 ≤ |x| < 1,
4.1
|x| ≥ 1,
where γ > 2, a ∈ C0, T , R, at > 0 for all t ∈ 0, T .
Obviously, hypotheses HF1 i, ii, and v are satisfied. Moreover,
|Ft, x| ≤
⎧
πγ α0
π γ−α0 α0
π
π γ
⎪
⎪
|x| ≤ at |x|α0 ≤ at
|x| ,
⎨at 2 |x| ≤ at 2 |x|
2
2
πγ
⎪
⎪
⎩at ≤ at
,
2
⎧
π
πγ
πγ γ−1
πγ
⎪
cos
,
|x|γ |x|γ−2 x ≤ at
|x| ≤ at
⎨at
2
2
2
2
|∇Ft, x| ⎪
⎩0,
0 ≤ |x| < 1,
|x| ≥ 1,
0 ≤ |x| < 1,
|x| ≥ 1.
4.2
Abstract and Applied Analysis
13
Therefore,
|∇Ft, x|, |Ft, x| ≤
πγat πγat α0
|x| .
2
2
4.3
So, condition HF1 iii holds.
On the other hand,
γπ/2at cos π/2|x|γ |x|γ
π
∇Ft, x, x
lim sup
lim sup
γat,
γ
γ
2
|x|
|x|
|x| → 0
|x| → 0
4.4
uniformly for a.a. t ∈ 0, T , so condition HF1 iv holds.
C We can find the following potential functions satisfying the conditions stated in
Theorem 3.4:
⎧
at γ
⎪
⎪
⎪
⎨− γ |x| ,
F1 t, x π
⎪ at
⎪
⎪
cos
|x|2 −
⎩
π
2
π
⎧
γ
⎪
⎪at sin 2 |x| ,
⎪
⎨
F2 t, x 1
1
⎪
⎪
at
|x|2 −
⎪
⎩
|x| 2
|x| < 1,
at
,
γ
|x| ≥ 1,
|x| < 1,
1
,
2
⎧
at γ
⎪
⎪
⎪
⎨ γ |x| ,
F3 t, x ⎪
2at
1
⎪
⎪
,
−
at
2
⎩
γ
|x|
|x| ≥ 1,
4.5
|x| < 1,
|x| ≥ 1,
where a ∈ C0, T , R, at > 0 for all t ∈ 0, T .
It is clear that Fi x, 0 0 i 1, 2, 3 for a.e. t ∈ 0, T , and hypotheses HF2 i and
HF2 ii are satisfied. A direct verification shows that conditions HF2 v and HF2 vi
are satisfied. Note that
⎧
γ−2 x
0 ≤ |x| < 1,
⎪
≤ at|x|γ−1 ≤ 2at,
−at|x|
⎪
⎨
|∇F1 t, x| ⎪−at sin π |x|2 x ≤ at|x| ≤ 2at|x|γ , |x| ≥ 1,
⎪
⎩
2
14
Abstract and Applied Analysis
⎧
π
πγ
π
γ π
γ−2 ⎪
⎪
, 0 ≤ |x| < 1,
x
cos
|x|
|x|
≤ γat |x|γ−1 ≤ at
γat
⎪
⎪
2
2
2
2
⎨
|∇F2 t, x| x 1
⎪
at x − 3 ≤ at |x| 2 ≤ at at|x|γ , |x| ≥ 1,
⎪
⎪
⎪
|x| |x|
⎩
⎧
γ−2 25/2 ⎪
≤ at,
⎨at|x| x ≤ at|x|
|∇F3 t, x| ⎪at|x|−5/2 x ≤ at|x| ≤ at|x|γ ,
⎩
0 ≤ |x| < 1,
|x| ≥ 1,
|F1 t, x| ≤ at|x|γ at1 1 2at 2at|x|γ ,
π
1
1
3at at|x|γ
,
≤
|x|γ at 1 |x|γ −
|F2 t, x| ≤ atsin
2
2
2
2
1
2at
1
≤ 5at at|x|γ .
at 2 |F3 t, x| ≤ at |x|γ γ
γ
|x|
4.6
So,
|∇F1 t, x|, |F1 t, x| ≤ 2at 1 |x|γ ,
πγ 1 |x|γ ,
2
|∇F3 t, x|, |F3 t, x| ≤ 5at 1 |x|γ ,
|∇F2 t, x|, |F2 t, x| ≤ at
−at|x|γ
∇F1 t, x, x
lim
−at,
|x| → 0
|x|γ
|x|γ
|x| → 0
at cos π/2|x|γ π/2γ|x|γ
∇F2 t, x, x
lim sup
lim
|x| → 0
|x|γ
|x|γ
|x| → 0
lim sup
lim at cos
|x| → 0
π
2
|x|γ
π
2
γ
4.7
γπat
,
2
γ
lim sup
|x| → 0
at|x|
∇F3 t, x, x
lim
at,
γ
|x| → 0 |x|
|x|γ
which shows that assumptions HF2 iii and HF2 iv are fulfilled.
Acknowlegments
This work was supported by the National Natural Science Foundation of China nos.
11126286, 11001063, and 11201095, the Fundamental Research Funds for the Central Universities no. 2012, China Postdoctoral Science Foundation funded project no. 20110491032,
and China Postdoctoral Science Special Foundation no. 2012T50325.
Abstract and Applied Analysis
15
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