Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations
Hindawi
Discrete Dynamics in Nature and Society
Volume 2017, Article ID 9281049, 9 pages
https://doi.org/10.1155/2017/9281049
Research Article
Nontrivial Solutions for Time Fractional Nonlinear
Schrödinger-Kirchhoff Type Equations
N. Nyamoradi,1 Y. Zhou,2,3 E. Tayyebi,1 B. Ahmad,3 and A. Alsaedi3
1
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
3
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University,
Jeddah 21589, Saudi Arabia
2
Correspondence should be addressed to Y. Zhou;
Received 4 April 2017; Revised 11 June 2017; Accepted 22 June 2017; Published 27 July 2017
Academic Editor: Thabet Abdeljawad
Copyright © 2017 N. Nyamoradi et al. 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 existence of solutions for time fractional Schrödinger-Kirchhoff type equation involving left and right Liouville-Weyl
fractional derivatives via variational methods.
1. Introduction
In recent years, there has been a great interest in studying
problems involving fractional Schrödinger equations [1–
5], Kirchhoff type equations [6–8], fractional Navier-Stokes
equations [9, 10], and fractional ordinary differential equations and Hamiltonian systems [11–17], and so forth. For
further details and applications, we refer the reader to [18, 19]
and the references cited therein.
On the other hand, the integer-order SchrödingerKirchhoff type equations have also been investigated by
many authors; for example, see [20–23]. In fact, SchrödingerKirchhoff type equations play an important role in modelling
several physical and biological systems. However, to the best
of our knowledge, the existence of solutions to the time
fractional Schrödinger-Kirchhoff type equations has yet to be
addressed.
The objective of the present paper is to study time
fractional Schrödinger-Kirchhoff type equation of the form
(𝑎 + 𝑏 ∫ −∞ 𝐷𝑡𝛼 𝑢 (𝑡) 𝑑𝑡)
2
R
+ 𝜇𝑉 (𝑡) 𝑢 = 𝑓 (𝑡, 𝑢) ,
𝜃−1
𝛼
𝛼
𝑡 𝐷∞ ( −∞ 𝐷𝑡 𝑢 (𝑡))
(1)
𝛼
𝑡 ∈ R, 𝑢 ∈ 𝐻 (R) ,
𝛼
where 𝛼 ∈ (1/2, 1], −∞ 𝐷𝑡𝛼 and 𝑡 𝐷∞
, respectively, denote left
and right Liouville-Weyl fractional derivatives of order 𝛼 on
R, 𝑎, 𝑏 > 0 are constants, 𝜇 > 0 is parameter, 𝜃 > 1, 𝑓 ∈
𝐶(R × R, R), and 𝑉 : R → R+ is a potential function.
The rest of the paper is organized as follows. Section 2
contains preliminary concepts of fractional calculus and
fractional Sobolev space, while some important lemmas,
which are needed in the proof of main results, are obtained
in Section 3. We present our main results in Section 4.
2. Preliminaries
In this section, we recall important definitions and concepts
of fractional calculus and then prove certain results about
fractional Sobolev space 𝐻𝛼 (R) related to our study of the
problem at hand.
Definition 1 (see [24]). The left and right Liouville-Weyl
fractional integrals of order 𝛼 ∈ (0, 1) on R are defined by
𝛼
−∞ 𝐼𝑥 𝜙 (𝑥) =
𝑥
1
∫ (𝑥 − 𝜉)𝛼−1 𝜙 (𝜉) 𝑑𝜉,
Γ (𝛼) −∞
𝛼
𝑥 𝐼∞ 𝜙 (𝑥) =
∞
1
∫ (𝜉 − 𝑥)𝛼−1 𝜙 (𝜉) 𝑑𝜉,
Γ (𝛼) 𝑥
respectively, where 𝑥 ∈ R.
(2)
2
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The left and right Liouville-Weyl fractional derivatives of
order 𝛼 ∈ (0, 1) on R are defined by
𝑑
𝐼1−𝛼 𝜙 (𝑥) ,
𝑑𝑥 −∞ 𝑥
𝛼
−∞ 𝐷𝑥 𝜙 (𝑥) =
𝑑 1−𝛼
𝛼
𝑥 𝐷∞ 𝜙 (𝑥) = −
𝑥 𝐼∞ 𝜙 (𝑥) ,
⟨𝑢, V⟩𝑋𝛼
(3)
𝛼
𝑥 𝐷∞ 𝜙 (𝑥) =
∞ 𝜙
(𝑥) − 𝜙 (𝑥 + 𝜉)
𝛼
𝑑𝜉.
∫
Γ (1 − 𝛼) 0
𝜉𝛼+1
‖𝑢‖2𝑋𝛼 = ⟨𝑢, 𝑢⟩𝑋𝛼 .
(13)
𝑋𝜇𝛼 = {𝑢 ∈ 𝑋𝛼 : ∫ 𝜇𝑉 (𝑡) |𝑢|2 𝑑𝑡 < +∞} ,
(14)
Define the space
R
(4)
with the norm
2
‖𝑢‖𝑋𝜇𝛼 = (∫ 𝑎𝜃−1 ( −∞ 𝐷𝑡𝛼 𝑢 (𝑡) ) 𝑑𝑡
R
Also, we define the Fourier transform F(𝑢)(𝜉) of 𝑢(𝑥) as
F (𝑢) (𝜉) = ∫
∞
−∞
2
𝑒−𝑖𝑥⋅𝜉 𝑢 (𝑥) 𝑑𝑥.
+ ∫ 𝜇𝑉 (𝑡) |𝑢| 𝑑𝑡)
(5)
For any 𝛼 > 0, we define the seminorm and norm,
respectively, as [16]
𝛼
= −∞ 𝐷𝑥𝛼 𝑢𝐿2 ,
|𝑢|𝐼−∞
1/2
𝛼
= (‖𝑢‖2𝐿2 + |𝑢|2𝐼−∞
‖𝑢‖𝐼−∞
𝛼 )
,
(6)
𝛼
(R) denote the completion of 𝐶0∞ (R)
and let the space 𝐼−∞
𝛼 .
with respect to the norm ‖ ⋅ ‖𝐼−∞
Next, for 0 < 𝛼 < 1, we give the relationship between
𝛼
classical fractional Sobolev space 𝐻𝛼 (R) and 𝐼−∞
(R), where
𝛼
𝐻 (R) is defined by
(15)
.
Lemma 2. (𝑋𝜇𝛼 , ‖ ⋅ ‖𝑋𝜇𝛼 ) is a uniformly convex Banach space.
Proof. 𝑋𝜇𝛼 is obviously Banach space. Now, we can prove that
(𝑋𝜇𝛼 , ‖ ⋅ ‖𝑋𝜇𝛼 ) is uniformly convex. To this end, let 0 < 𝜀 < 2
and 𝑢, V ∈ 𝑋𝜇𝛼 with ‖𝑢‖𝑋𝜆𝛼 = ‖V‖𝑋𝜇𝛼 = 1 and ‖𝑢 − V‖𝑋𝜇𝛼 ≥ 𝜀.
Using the following inequality:
𝑎 + 𝑏 2 𝑎 − 𝑏 2 1
2
2
+
≤ (|𝑎| + |𝑏| ) , ∀𝑎, 𝑏 ∈ R,
2 2
2
we get
1/2
‖𝑢‖𝛼 = (‖𝑢‖2𝐿2 + |𝑢|2𝛼 ) ,
(8)
𝛼
|𝑢|𝛼 = 𝜉 F (𝑢)𝐿2 .
𝑢 + V 2
𝑑𝑡
+ ∫ 𝜇𝑉 (𝑡)
2
R
2
𝑢−V
+ ∫ 𝑎𝜃−1 ( −∞ 𝐷𝑡𝛼 (
) (𝑡) ) 𝑑𝑡
2
R
(9)
𝑢 − V 2
𝑑𝑡
+ ∫ 𝜇𝑉 (𝑡)
2
R
,
and seminorm
𝛼
(R) are equal and
Observe that the spaces 𝐻𝛼 (R) and 𝐼−∞
have equivalent norms (see [16]).
Therefore, we define
𝛼
𝐻𝛼 (R) = {𝑢 ∈ 𝐿2 (R) | 𝜉 F (𝑢) ∈ 𝐿2 (R)} .
(10)
Let
≤
R
2
+ ∫ 𝑎𝜃−1 ( −∞ 𝐷𝑡𝛼 V (𝑡) ) 𝑑𝑡 + ∫ 𝜇𝑉 (𝑡) |𝑢|2 𝑑𝑡
R
R
(11)
(17)
1
2
(∫ 𝑎𝜃−1 ( −∞ 𝐷𝑡𝛼 𝑢 (𝑡) ) 𝑑𝑡
2 R
+ ∫ 𝜇𝑉 (𝑡) |V|2 𝑑𝑡) =
2
𝑋𝛼 = {𝑢 ∈ 𝐻𝛼 (R) | ∫ ( −∞ 𝐷𝑡𝛼 𝑢 (𝑡) + |𝑢 (𝑡)|2 ) 𝑑𝑡
(16)
𝑢 + V 2
2
+ 𝑢 − V
𝛼
2 𝑋𝜇 2 𝑋𝜇𝛼
2
𝑢+V
= ∫ 𝑎𝜃−1 ( −∞ 𝐷𝑡𝛼 (
) (𝑡) ) 𝑑𝑡
2
R
‖⋅‖𝛼
with the norm
< ∞} .
R
1/2
(7)
𝐻𝛼 (R) = 𝐶0∞ (R)
(12)
and the corresponding norm
respectively, where 𝑥 ∈ R.
The definitions (3) may be written in an alternative form
as follows:
∞ 𝜙
(𝑥) − 𝜙 (𝑥 − 𝜉)
𝛼
𝑑𝜉,
∫
Γ (1 − 𝛼) 0
𝜉𝛼+1
= ∫ ( −∞ 𝐷𝑡𝛼 𝑢 (𝑡) ⋅ −∞ 𝐷𝑡𝛼 V (𝑡) + 𝑢 (𝑡) V (𝑡)) 𝑑𝑡
R
𝑑𝑥
𝛼
−∞ 𝐷𝑥 𝜙 (𝑥) =
The space 𝑋𝛼 is a reflexive and separable Hilbert space with
the inner product
R
1
(‖𝑢‖2𝑋𝜇𝛼 + ‖V‖2𝑋𝜇𝛼 ) = 1,
2
which implies that ‖(𝑢 + V)/2‖2𝑋𝜇𝛼 ≤ 1 − 𝜀/2. Hence, taking
𝛿 = 𝛿(𝜀) such that 1−𝜀/2 = 1−𝛿, we have ‖(𝑢+V)/2‖2𝑋𝜇𝛼 ≤ 1−𝛿.
Therefore, (𝑋𝜇𝛼 , ‖ ⋅ ‖𝑋𝜇𝛼 ) is uniformly convex.
Discrete Dynamics in Nature and Society
3
In the sequel, we need the following assumptions.
Proof. The proof is similar to that of Theorem 2.1 in [16], so
we omit it.
(V1) 𝑉(𝑡) ∈ 𝐶(R, R), 𝑉0 fl inf 𝑡∈R 𝑉(𝑡) > 0;
(V2) there exists 𝑟 > 0 such that, for any 𝑀 > 0,
Also by Lemma 4, there is a constant 𝐶𝛼 > 0 such that
meas ({𝑡 ∈ (𝑦 − 𝑟, 𝑦 + 𝑟) : 𝑉 (𝑡) ≤ 𝑀}) → 0
‖𝑢‖∞ ≤ 𝐶𝛼 ‖𝑢‖𝑋𝜇𝛼 .
(18)
as 𝑦 → ∞;
Remark 5. If 𝑢 ∈ 𝐻𝛼 (R) with 1/2 < 𝛼 < 1, then it follows by
Lemma 4 that 𝑢 ∈ 𝐿𝑞 (R) for all 𝑞 ∈ [2, ∞) as
(V3) there exists 𝑙0 > 0 such that ∫|𝑡|≥𝑙 𝑉(𝑡)−1 𝑑𝑡 < ∞;
2
∫ |𝑢 (𝑥)|𝑞 𝑑𝑥 ≤ ‖𝑢‖𝑞−2
∞ ‖𝑢‖𝐿2 (R) .
0
(F1) 𝑓 ∈ 𝐶(R × R, R) and there exist constants 𝑐0 , 𝑐1 ,
. . . , 𝑐𝑙 > 0 and 𝑞𝑗 ∈ (2, 2𝜃) such that
𝑙
𝑞 −1
𝑓 (𝑡, 𝑢) ≤ 𝑐0 |𝑢| + ∑ 𝑐𝑗 |𝑢| 𝑗 , ∀ (𝑡, 𝑢) ∈ R × R;
(19)
𝑗=1
Remark 6. From Remark 5 and Lemma 3, it is easy to verify
that the imbedding of 𝑋𝜇𝛼 in 𝐿𝑞 (R) is also compact for 𝑞 ∈
(2, ∞). H (...truncated)
