Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 842594, 6 pages
http://dx.doi.org/10.1155/2013/842594
Research Article
Standing Wave Solutions for the Discrete Coupled Nonlinear
Schrödinger Equations with Unbounded Potentials
Meihua Huang1,2 and Zhan Zhou1,2
1
2
School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China
Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes,
Guangzhou University, Guangzhou, Guangdong 510006, China
Correspondence should be addressed to Zhan Zhou; zzhou0321@hotmail.com
Received 18 January 2013; Accepted 25 February 2013
Academic Editor: Chuangxia Huang
Copyright © 2013 M. Huang and Z. Zhou. 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 demonstrate the existence of standing wave solutions of the discrete coupled nonlinear Schrödinger equations with unbounded
potentials by using the Nehari manifold approach and the compact embedding theorem. Sufficient conditions are established to
show that the standing wave solutions have both of the components not identically zero.
where the amplitude 𝜙𝑛 and 𝜓𝑛 are supposed to be real.
Inserting the ansatz of the standing wave solutions (3) into
(1), we obtain the following equivalent algebraic equations:
1. Introduction
Consider the coupled discrete Schrödinger system
𝑖
𝑑𝑢𝑛
󵄨 󵄨2
󵄨 󵄨2
= −(A𝑢)𝑛 + 𝑏1𝑛 𝑢𝑛 − 𝑎1 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑢𝑛 − 𝑎3 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 𝑢𝑛 ,
𝑑𝑡
𝑑V
󵄨 󵄨2
󵄨 󵄨2
𝑖 𝑛 = −(AV)𝑛 + 𝑏2𝑛 V𝑛 − 𝑎2 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 V𝑛 − 𝑎3 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 V𝑛 ,
𝑑𝑡
󵄨 󵄨2
󵄨 󵄨2
−(A𝜙)𝑛 − 𝜔1 𝜙𝑛 + 𝑏1𝑛 𝜙𝑛 − 𝑎1 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨 𝜙𝑛 − 𝑎3 󵄨󵄨󵄨𝜓𝑛 󵄨󵄨󵄨 𝜙𝑛 = 0,
(1)
where 𝑎𝑖 > 0, {𝑏𝑗𝑛 } are real valued sequences, 𝑖 =
1, 2, 3, and 𝑗 = 1, 2. A is the discrete Laplacian operator
defined as (A𝑢)𝑛 = 𝑢𝑛+1 + 𝑢𝑛−1 − 2𝑢𝑛 .
The system (1) could be viewed as the discretization of the
two-component system of time-dependent nonlinear GrossPitaevskii system (see [1] for detail)
𝑖ℎ𝜕𝑡 𝑢 = −
ℎ2
Δ𝑢 + 𝑏1 (𝑥) 𝑢 − 𝑎1 |𝑢|2 𝑢 − 𝑎3 |V|2 𝑢,
2𝑚
ℎ2
𝑖ℎ𝜕𝑡 V = −
ΔV + 𝑏2 (𝑥) V − 𝑎2 |V|2 V − 𝑎3 |𝑢|2 V.
2𝑚
(2)
In this paper, we will study the standing wave solutions of
(1), that is, solutions of the form
𝑢𝑛 = exp (−𝑖𝜔1 𝑡) 𝜙𝑛 ,
V𝑛 = exp (−𝑖𝜔2 𝑡) 𝜓𝑛 ,
𝑛 ∈ Z, (3)
󵄨 󵄨2
󵄨 󵄨2
−(A𝜓)𝑛 − 𝜔2 𝜓𝑛 + 𝑏2𝑛 𝜓𝑛 − 𝑎2 󵄨󵄨󵄨𝜓𝑛 󵄨󵄨󵄨 𝜓𝑛 − 𝑎3 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨 𝜓𝑛 = 0.
(4)
Since Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was
realized in 1997 (see [2]), this stimulated various analytical
and numerical results on the standing wave solutions of the
system (2). The discrete nonlinear Schrödinger equations
(DNLS) have a crucial role in the modeling of a great variety
of phenomena, ranging from solid-state and condensedmatter physics to biology. During the last years, there has
been a growing interest in approaches to the existence
problem for standing waves. We refer to the continuation
methods in [3, 4], which have been proved powerful for
both theoretical considerations and numerical computations
(see [5]), to [6], which exploits spatial dynamics and centre
manifold reduction, and to the variational methods in [7–11],
which rely on critical point techniques (linking theorems and
Nehari manifold).
We noticed that most works on the existence of
standing waves solutions are for single discrete nonlinear
2
Abstract and Applied Analysis
Schrödinger equation, and less is known for discrete nonlinear Schrödinger system. In the recent paper [12], the authors
considered the standing wave solutions of the following
system:
𝑖
𝑑𝑢𝑛
󵄨 󵄨2
󵄨 󵄨2
= −(A𝑢)𝑛 + 𝑏1𝑛 𝑢𝑛 − 𝑐1𝑛 V𝑛 − 𝑎1 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑢𝑛 − 𝑎3 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 𝑢𝑛 ,
𝑑𝑡
𝑑V
󵄨 󵄨2
󵄨 󵄨2
𝑖 𝑛 = −(AV)𝑛 + 𝑏2𝑛 V𝑛 − 𝑐2𝑛 𝑢𝑛 − 𝑎2 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 V𝑛 − 𝑎3 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 V𝑛 ,
𝑑𝑡
Without loss of generality we assume that 𝑉𝑖 ≥ 1, 𝑖 = 1, 2;
that is 𝑏𝑖𝑛 ≥ 1 for 𝑛 ∈ Z, 𝑖 = 1, 2. Let
𝐿 𝑖 = −A + 𝑉𝑖 ,
𝑖 = 1, 2.
1/𝑝
< ∞,
Lemma 1. If 𝑉𝑖 , 𝑖 = 1, 2, satisfy the condition (10), then for
any 2 ≤ 𝑝 ≤ ∞, 𝐸1 and 𝐸2 are compactly embedded into 𝑙𝑝 and
denote the best embedding constant 𝛼𝑝 = max‖𝜙‖𝑙𝑝 =1 1/‖𝜙‖𝐸1
and 𝛽𝑝 = max‖𝜙‖𝑙𝑝 =1 1/‖𝜙‖𝐸2 , respectively. Furthermore, the
spectra 𝜎(𝐿 1 ) and 𝜎(𝐿 2 ) are discrete, respectively.
By (11), (4) becomes
󵄨 󵄨2
󵄨 󵄨2
𝐿 1 𝜙𝑛 − 𝜔1 𝜙𝑛 − 𝑎1 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨 𝜙𝑛 − 𝑎3 󵄨󵄨󵄨𝜓𝑛 󵄨󵄨󵄨 𝜙𝑛 = 0,
󵄨 󵄨2
󵄨 󵄨2
𝐿 2 𝜓𝑛 − 𝜔2 𝜓𝑛 − 𝑎2 󵄨󵄨󵄨𝜓𝑛 󵄨󵄨󵄨 𝜓𝑛 − 𝑎3 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨 𝜓𝑛 = 0.
2
𝜙, 𝜓 ∈ 𝑙 .
(8)
Let us point out that the spectrum of −A in 𝑙2 coincides with
the interval [0, 4]. Obviously, we have
󵄩 󵄩2
0 ≤ (−A𝜙, 𝜙) ≤ 4󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 ,
∀𝜙 ∈ 𝑙2 .
(9)
Assume that the potential 𝑉𝑖 = {𝑏𝑖𝑛 }𝑛∈Z , 𝑖 = 1, 2, satisfies
lim 𝑏𝑖𝑛 = ∞,
1
∑ (𝑎 𝜙4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 ) .
4 𝑛∈Z 1 𝑛
𝑖 = 1, 2.
(10)
(14)
By Lemma 1, it follows that the action functional 𝐽(𝜙, 𝜓) ∈
𝐶1 (𝐸1 × 𝐸2 , R) and (13) corresponds to 𝐽󸀠 (𝜙, 𝜓) = 0. So we
define
𝐼 (𝜙, 𝜓) = (𝐽󸀠 (𝜙, 𝜓) , (𝜙, 𝜓))
= ((𝐿 1 − 𝜔1 ) 𝜙, 𝜙) + ((𝐿 2 − 𝜔2 ) 𝜓, 𝜓)
𝑛∈Z
|𝑛| → ∞
1
1
((𝐿 1 − 𝜔1 ) 𝜙, 𝜙) + ((𝐿 2 − 𝜔2 ) 𝜓, 𝜓)
2
2
−
}
𝜙𝑛 ∈ R, ∀𝑛 ∈ Z} .
}
For the case 𝑝 = 2, we need the usual Hilbert space of 𝑙2 ,
endowed with the real scalar product
(13)
Now we can define the action functional
(6)
Between 𝑙𝑝 spaces the following elementary embedding
relation holds:
󵄩 󵄩
󵄩 󵄩
𝑙𝑞 ⊂ 𝑙𝑝 , 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙𝑝 ≤ 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙𝑞 , 1 ≤ 𝑞 ≤ 𝑝 ≤ ∞.
(7)
(12)
The following lemma can be found in [9].
𝐽 (𝜙, 𝜓) =
(𝜙, 𝜓) = ∑ 𝜙𝑛 𝜓𝑛 ,
󵄩 1/2 󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜙󵄩󵄩𝐸𝑖 = 󵄩󵄩󵄩󵄩𝐿 𝑖 𝜙󵄩󵄩󵄩󵄩𝑙2 ,
2
𝐸𝑖 = {𝜙 ∈ 𝑙2 : 𝐿1/2
𝑖 𝜙 ∈ 𝑙 },
2. Preliminaries
In this section we describe the functional setting needed for
the treatment of the infinite nonlinear system (4). We first
introduce a compact embedding theorem.
Consider the real sequence spaces
(11)
which are self-adjoint operators defined on 𝑙2 , and
(5)
which is more general than the system (1). However, they
make a mistake to obtain the equivalent algebraic equations
because 𝜔1 may be different from 𝜔2 . Hence, there are two
ways to correct this mistake. The first method is to study the
special standing wave solutions (3) of the system (5) with 𝜔1 =
𝜔2 . The second method is to study the standing wave solutions
(3) of the system (5) with 𝑐1𝑛 ≡ 0 and 𝑐2𝑛 ≡ 0, 𝑛 ∈ Z. In this
paper, we consider the second method. By the way, the proof
of the main results in [12] is also not fully corrected.
The paper is organized as follows. In Section 2, we
introduce some preliminaries and a discrete version of compact embedding theorem. Some key lemmas on the Nehari
manifold are proved in Section 3. In Section 4, the main
results are stated and proved.
{
󵄨 󵄨𝑝
󵄩 󵄩
𝑙𝑝 = {𝜙 = {𝜙𝑛 }𝑛∈Z : 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙𝑝 = ( ∑ 󵄨󵄨󵄨𝜙𝑛 󵄨󵄨󵄨 )
𝑛∈Z
{
𝑖 = 1, 2,
−∑
𝑛∈Z
(𝑎1 𝜙𝑛4
+
𝑎2 𝜓𝑛4
+
(15)
2𝑎3 𝜙𝑛2 𝜓𝑛2 ) ,
and the Nehari manifold
𝑁 = {(𝜙, 𝜓) ∈ 𝐸1 × 𝐸2 : 𝐼 (𝜙, 𝜓) = 0, (𝜙, 𝜓) ≠ 0} .
(16)
3. Some Lemmas on the Nehari Manifold
Let
𝜆 𝑖 = inf {𝜎 (𝐿 𝑖 )} ,
𝑖 = 1, 2.
(17)
To prove the main results, we need some lemmas on the
Nehari manifold.
Lemma 2. Assume that 𝜔1 < 𝜆 1 , 𝜔2 < 𝜆 2 , and (10) holds.
Then the Nehari manifold 𝑁 is nonempty in 𝐸1 × 𝐸2 . Furthermore, for (𝜙, 𝜓) ∈ 𝑁, 𝐽(𝑡𝜙, 𝑡𝜓) attains a unique maximum
point at 𝑡 = 1.
Abstract and Applied Analysis
3
𝛼𝑝 and 𝛽𝑝 , we get 𝜆 1 = 1/𝛼22 and 𝜆 2 = 1/𝛽22 . For any (𝜙, 𝜓) ∈
𝑁, we have
Proof. First we show that 𝑁 ≠ 0.
From (15) and (16), we rewrite
𝐽 (𝜙, 𝜓) =
󵄩 󵄩2 󵄩 󵄩 2
󵄩 󵄩2
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜙󵄩󵄩𝐸1 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2
1 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
(󵄩𝜙󵄩 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 )
2 󵄩 󵄩𝐸1
= ∑ (𝑎1 𝜙𝑛4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 )
𝑛∈Z
1 󵄩 󵄩2
󵄩 󵄩2
+ (󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2 )
2
−
󵄩 󵄩 2 󵄩 󵄩2 2
≤ 𝑎∗ (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙4 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙4 )
(18)
1
∑ (𝑎 𝜙4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 ) ,
4 𝑛∈Z 1 𝑛
󵄩 󵄩2
󵄩 󵄩2 2
≤ 𝑎∗ (𝛼42 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 + 𝛽42 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 )
󵄩 󵄩2 󵄩 󵄩 2 2
≤ 𝑎∗ 𝛾22 (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 ) ,
󵄩 󵄩2 󵄩 󵄩 2
󵄩 󵄩2
󵄩 󵄩2
𝐼 (𝜙, 𝜓) = 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2
− ∑ (𝑎1 𝜙𝑛4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 ) .
(22)
(19)
𝑛∈Z
where 𝑎∗ = max {𝑎1 , 𝑎2 , 𝑎3 } and 𝛾2 = max{𝛼42 , 𝛽42 }.
Let
𝛾1 = min {1, 1 −
Let (𝜙, 𝜓) ∈ (𝐸1 − {0}) × (𝐸2 − {0}); then by (19)
𝜔1
𝜔
,1 − 2}.
𝜆1
𝜆2
(23)
By (22), it is easy to see that
󵄩 󵄩2 󵄩 󵄩 2
󵄩 󵄩2 󵄩 󵄩 2 2
𝛾1 (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 ) ≤ 𝑎∗ 𝛾22 (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 ) ,
󵄩 󵄩2
󵄩 󵄩2
𝐼 (𝑡𝜙, 𝑡𝜓) = 𝑡2 (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 )
󵄩 󵄩2
󵄩 󵄩2
+ 𝑡2 (󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2 )
and this implies that
𝛾
󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜙󵄩󵄩𝐸1 + 󵄩󵄩𝜓󵄩󵄩𝐸2 ≥ ∗1 2 .
𝑎 𝛾2
− 𝑡4 ∑ (𝑎1 𝜙𝑛4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 )
𝑛∈Z
(20)
󵄩 󵄩2 󵄩 󵄩 2
󵄩 󵄩2
󵄩 󵄩2
= 𝑡2 (󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸1 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2
𝑛∈Z
Notice that ‖𝜙‖2𝐸1 − 𝜔1 ‖𝜙‖2𝑙2 ≥ (𝜆 1 − 𝜔1 )‖𝜙‖2𝑙2 > 0 and
‖𝜓‖2𝐸2 − 𝜔2 ‖𝜓‖2𝑙2 ≥ (𝜆 2 − 𝜔2 )‖𝜓‖2𝑙2 > 0; by (20), we see that
𝐼(𝑡𝜙, 𝑡𝜓) > 0 for 𝑡 > 0 small enough and 𝐼(𝑡𝜙, 𝑡𝜓) < 0 for
𝑡 > 0 large enough. As a consequence, there exists 𝑡0 > 0 such
that 𝐼(𝑡0 𝜙, 𝑡0 𝜓) = 0; that is, (𝑡0 𝜙, 𝑡0 𝜓) ∈ 𝑁.
Let 𝐹(𝑡) = 𝐽(𝑡𝜙, 𝑡𝜓), (𝜙, 𝜓) ∈ 𝑁. Computing the
derivative of 𝐹, we have
2
𝐹 (𝑡) = 𝑡 (1 − 𝑡 ) ∑
𝑛∈Z
(𝑎1 𝜙𝑛4
+
𝑎2 𝜓𝑛4
+
2𝑎3 𝜙𝑛2 𝜓𝑛2 ) .
(25)
Moreover, we have
1
𝐽 (𝜙, 𝜓) = 𝐽 (𝜙, 𝜓) − 𝐼 (𝜙, 𝜓)
4
−𝑡2 ∑ (𝑎1 𝜙𝑛4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 )) .
󸀠
(24)
(21)
This shows that 𝑡 = 1 is a unique maximum point. The proof
is completed.
Lemma 3. Assume that 𝜔1 < 𝜆 1 , 𝜔2 < 𝜆 2 , and (10) holds.
Then there exists 𝜂 > 0 such that 𝐽(𝜙, 𝜓) ≥ 𝜂, for all (𝜙, 𝜓) ∈ 𝑁.
Proof. Since 𝜆 1 is the smallest eigenvalue of 𝐸1 and 𝜆 2 is the
smallest eigenvalue of 𝐸2 , from the definition of the constant
=
1 󵄩󵄩 󵄩󵄩2
󵄩 󵄩 2 󵄩 󵄩2
󵄩 󵄩2
(󵄩𝜙󵄩 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸2 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2 )
4 󵄩 󵄩𝐸1
≥
𝛾2
𝛾1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
(󵄩󵄩𝜙󵄩󵄩𝐸1 + 󵄩󵄩𝜓󵄩󵄩𝐸2 ) ≥ ∗1 2 .
4
4𝑎 𝛾2
(26)
Let 𝜂 = 𝛾12 /(4𝑎∗ 𝛾22 ); then we get 𝐽(𝜙, 𝜓) ≥ 𝜂, for all (𝜙, 𝜓) ∈ 𝑁.
The proof is completed.
4. Main Results
Now we state our main results in this paper as follows.
Theorem 4. Assume that 𝜔1 < 𝜆 1 , 𝜔2 < 𝜆 2 , and (10) holds.
Then system (13) has a nontrivial solution in 𝐸1 × 𝐸2 ; that is,
system (1) has a nontrivial standing wave solution.
In order to prove Theorem 4, we consider the following
constrained minimization problem:
𝑑≡
inf 𝐽 (𝜙, 𝜓) .
(𝜙,𝜓)∈𝑁
(27)
From the standard variational method, the proof of
Theorem 4 is changed into finding a solution to the
minimization problem (27). Now we are ready to prove
Theorem 4.
4
Abstract and Applied Analysis
Proof. Let 𝑑 be given by (27). By Lemma 2, 𝑁 is nonempty
and there exists a sequence {(𝜙(𝑘) , 𝜓(𝑘) )} ⊂ 𝑁 such that
𝑑 = lim 𝐽 (𝜙(𝑘) , 𝜓(𝑘) ) .
(28)
𝑘→∞
By Lemma 3, 𝑑 > 0 and 𝑑 ≤ 𝐷 = max𝑘 {𝐽(𝜙(𝑘) , 𝜓(𝑘) )} <
∞. By virtue of (26), we have
4
4𝐷
󵄩󵄩 (𝑘) 󵄩󵄩2 󵄩󵄩 (𝑘) 󵄩󵄩2
󵄩󵄩𝜙 󵄩󵄩 + 󵄩󵄩𝜓 󵄩󵄩 ≤ 𝐽 (𝜙(𝑘) , 𝜓(𝑘) ) ≤
< ∞.
󵄩𝐸2 𝛾1
󵄩 󵄩𝐸1 󵄩
𝛾1
(29)
Thus, sequences {𝜙(𝑘) } and {𝜓(𝑘) } are bounded in Hilbert
spaces 𝐸1 and 𝐸2 , respectively. Therefore, there exist subsequences of {𝜙(𝑘) } and {𝜓(𝑘) } (denoted by itself) that weakly
converge to some 𝜙∗ ∈ 𝐸1 and 𝜓∗ ∈ 𝐸2 , respectively. By
Lemma 1, we get, for any 2 ≤ 𝑝 ≤ ∞,
lim 𝜙(𝑘) = 𝜙∗ ,
𝑘→∞
lim 𝜓(𝑘) = 𝜓∗ ,
in 𝑙𝑝 .
𝑘→∞
Next, we show that (𝜙∗ , 𝜓∗ ) ∈ 𝑁 and 𝐽(𝜙∗ , 𝜓∗ ) = 𝑑.
Since 𝐸1 and 𝐸2 are Hilbert spaces, by (32) we have
󵄩󵄩 ∗ 󵄩󵄩2 󵄩󵄩 ∗ 󵄩󵄩2
󵄩󵄩𝜙 󵄩󵄩𝐸1 + 󵄩󵄩𝜓 󵄩󵄩𝐸2
󵄩󵄩
󵄩󵄩2 󵄩󵄩
󵄩󵄩2
= 󵄩󵄩󵄩󵄩weak − lim 𝜙(𝑘) 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩weak − lim 𝜓(𝑘) 󵄩󵄩󵄩󵄩
𝑘→∞
𝑘→∞
󵄩
󵄩𝐸1 󵄩
󵄩𝐸2
󵄩 󵄩2
󵄩
󵄩2
≤ lim inf 󵄩󵄩󵄩󵄩𝜙(𝑘) 󵄩󵄩󵄩󵄩𝐸 + lim inf 󵄩󵄩󵄩󵄩𝜓(𝑘) 󵄩󵄩󵄩󵄩𝐸
𝑘→∞
2
4
𝑘→∞
4
(35)
𝑛∈Z
2
2
+2𝑎3 (𝜙𝑛(𝑘) ) (𝜓𝑛(𝑘) ) )
(30)
󵄩 󵄩2
󵄩
󵄩2
+𝜔1 󵄩󵄩󵄩󵄩𝜙(𝑘) 󵄩󵄩󵄩󵄩𝑙 + 𝜔2 󵄩󵄩󵄩󵄩𝜓(𝑘) 󵄩󵄩󵄩󵄩𝑙 )
2
4
2
4
2
2
= ∑ (𝑎1 (𝜙𝑛∗ ) + 𝑎2 (𝜓𝑛∗ ) + 2𝑎3 (𝜙𝑛∗ ) (𝜓𝑛∗ ) )
𝐽 (𝜙(𝑘) , 𝜓(𝑘) )
𝑛∈Z
1 󵄩 󵄩2
1 󵄩
󵄩 󵄩2
󵄩2
󵄩
󵄩2
= (󵄩󵄩󵄩󵄩𝜙(𝑘) 󵄩󵄩󵄩󵄩𝐸 − 𝜔1 󵄩󵄩󵄩󵄩𝜙(𝑘) 󵄩󵄩󵄩󵄩𝑙2 ) + (󵄩󵄩󵄩󵄩𝜓(𝑘) 󵄩󵄩󵄩󵄩𝐸 − 𝜔2 󵄩󵄩󵄩󵄩𝜓(𝑘) 󵄩󵄩󵄩󵄩𝑙2 )
1
2
2
2
=
𝑘→∞
= lim inf ( ∑ (𝑎1 (𝜙𝑛(𝑘) ) + 𝑎2 (𝜓𝑛(𝑘) )
By virtue of (15) and (16), we have
4
4
2
2
1
∑ (𝑎1 (𝜙𝑛(𝑘) ) + 𝑎2 (𝜓𝑛(𝑘) ) + 2𝑎3 (𝜙𝑛(𝑘) ) (𝜓𝑛(𝑘) ) )
4 𝑛∈Z
−
1
󵄩2
󵄩 󵄩2 󵄩
≤ lim inf (󵄩󵄩󵄩󵄩𝜙(𝑘) 󵄩󵄩󵄩󵄩𝐸 + 󵄩󵄩󵄩󵄩𝜓(𝑘) 󵄩󵄩󵄩󵄩𝐸 )
1
2
𝑘→∞
4
4
2
2
1
∑ (𝑎 (𝜙(𝑘) ) + 𝑎2 (𝜓𝑛(𝑘) ) + 2𝑎3 (𝜙𝑛(𝑘) ) (𝜓𝑛(𝑘) ) ) .
4 𝑛∈Z 1 𝑛
(31)
First, we claim that
4
4
2
4
4
2
2
= ∑ (𝑎1 (𝜙𝑛∗ ) + 𝑎2 (𝜓𝑛∗ ) + 2𝑎3 (𝜙𝑛∗ ) (𝜓𝑛∗ ) ) .
(32)
=
𝑛∈Z
󵄨
󵄨󵄨
󵄨
2
+ ∑ 󵄨󵄨󵄨󵄨𝜓𝑛(𝑘) − 𝜓𝑛∗ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜓𝑛(𝑘) + 𝜓𝑛∗ 󵄨󵄨󵄨󵄨 (𝜙𝑛∗ ) .
2
𝜔2 ‖𝜓∗ ‖𝑙2 − ∑𝑛∈Z (𝑎1 (𝜙𝑛∗ )4 + 𝑎2 (𝜓𝑛∗ )4 + 2𝑎3 (𝜙𝑛∗ )2 (𝜓𝑛∗ )2 ) ≤ 0.
Through a similar argument to the proof of Lemma 2, we
know that 𝐼(𝑡𝜙∗ , 𝑡𝜓∗ ) is positive as 𝑡 is small enough. Therefore there exists 𝑡∗ ∈ (0, 1] such that 𝐼(𝑡∗ 𝜙∗ , 𝑡∗ 𝜓∗ ) = 0 which
implies (𝑡∗ 𝜙∗ , 𝑡∗ 𝜓∗ ) ∈ 𝑁. Thus we have 𝐽(𝑡∗ 𝜙∗ , 𝑡∗ 𝜓∗ ) =
(1/4)𝑊(𝑡∗ ) and by (32), 𝑊(1) = 4𝑑, where
4
2
2
𝑛∈Z
1
1
𝑑 ≤ 𝐽 (𝑡∗ 𝜙∗ , 𝑡∗ 𝜓∗ ) = 𝑊 (𝑡∗ ) ≤ 𝑊 (1) = 𝑑.
4
4
(33)
In fact,
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 ∑ (𝜙(𝑘) )2 (𝜓(𝑘) )2 − ∑ (𝜙∗ )2 (𝜓∗ )2 󵄨󵄨󵄨
𝑛
𝑛
𝑛 󵄨󵄨
󵄨󵄨󵄨𝑛∈Z 𝑛
󵄨󵄨
𝑛∈Z
󵄨
2
󵄨
󵄨󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨𝜙𝑛(𝑘) − 𝜙𝑛∗ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙𝑛(𝑘) + 𝜙𝑛∗ 󵄨󵄨󵄨󵄨 (𝜓𝑛(𝑘) )
2
Clearly, 𝑊(𝑡) is strictly increasing on 0 < 𝑡 < ∞. Therefore
by (27),
According to (30), it suffices to show that
2
2
∑ (𝜙𝑛∗ ) (𝜓𝑛∗ ) .
𝑛∈Z
2
𝑊 (𝑡) = 𝑡4 ∑ (𝑎1 (𝜙𝑛∗ ) + 𝑎2 (𝜓𝑛∗ ) + 2𝑎3 (𝜙𝑛∗ ) (𝜓𝑛∗ ) ) . (36)
𝑛∈Z
2
2
lim ∑ (𝜙𝑛(𝑘) ) (𝜓𝑛(𝑘) )
𝑘→∞
𝑛∈Z
2
which implies 𝐼(𝜙∗ , 𝜓∗ ) = ‖𝜙∗ ‖𝐸1 − 𝜔1 ‖𝜙∗ ‖𝑙2 + ‖𝜓∗ ‖𝐸2 −
4
2
lim ∑ (𝑎1 (𝜙𝑛(𝑘) ) + 𝑎2 (𝜓𝑛(𝑘) ) + 2𝑎3 (𝜙𝑛(𝑘) ) (𝜓𝑛(𝑘) ) )
𝑘→∞
𝑛∈Z
󵄩 󵄩2
󵄩 󵄩2
+ 𝜔1 󵄩󵄩󵄩𝜙∗ 󵄩󵄩󵄩𝑙2 + 𝜔2 󵄩󵄩󵄩𝜓∗ 󵄩󵄩󵄩𝑙2 ,
(34)
(37)
This implies that 𝑡∗ = 1 and 𝐽(𝜙∗ , 𝜓∗ ) = 𝑑.
Finally, we will prove (𝜙∗ , 𝜓∗ ) is a nontrivial solution to
system (13).
Since (𝜙∗ , 𝜓∗ ) is an energy minimizer on Nehari manifold
𝑁, there exists a Lagrange multiplier Λ such that
(𝐽󸀠 (𝜙∗ , 𝜓∗ ) + Λ𝐼󸀠 (𝜙∗ , 𝜓∗ ) , (𝜙, 𝜓)) = 0,
(38)
for any (𝜙, 𝜓) ∈ 𝐸1 × 𝐸2 . Let (𝜙, 𝜓) = (𝜙∗ , 𝜓∗ ) in (38).
(𝐽󸀠 (𝜙∗ , 𝜓∗ ), (𝜙∗ , 𝜓∗ )) = 𝐼(𝜙∗ , 𝜓∗ ) = 0 implies that
𝑛∈Z
Thus Hölder inequality and (30) imply the (33) holds.
Λ (𝐼󸀠 (𝜙∗ , 𝜓∗ ) , (𝜙∗ , 𝜓∗ )) = 0,
(39)
Abstract and Applied Analysis
5
̃ 𝑡∗ 𝜖𝜙)
̃
By (20) and 𝐼(𝑡∗ 𝜙,
̃
̃
̃
̃
(𝐻1 (𝜙, 𝜖𝜙))/(𝐻2 (𝜙, 𝜖𝜙)), where
but
(𝐼󸀠 (𝜙∗ , 𝜓∗ ) , (𝜙∗ , 𝜓∗ ))
− 4∑
𝑛∈Z
= −2 ∑
𝑛∈Z
4
(𝑎1 (𝜙𝑛∗ )
+
+
4
𝑎2 (𝜓𝑛∗ )
4
𝑎2 (𝜓𝑛∗ )
+
+
𝐻2 (𝜙, 𝜓) = ∑ (𝑎1 𝜙𝑛4 + 𝑎2 𝜓𝑛4 + 2𝑎3 𝜙𝑛2 𝜓𝑛2 ) ,
2
2
2𝑎3 (𝜙𝑛∗ ) (𝜓𝑛∗ ) )
2
2
2𝑎3 (𝜙𝑛∗ ) (𝜓𝑛∗ ) )
< 0.
∗̃
̃ = (𝐻2 (𝜙,
̃ 𝜖𝜙))/(4𝐻
̃
̃ ̃
and 𝐽(𝑡 𝜙, 𝑡 𝜖𝜙)
2 (𝜙, 𝜖𝜙)).
1
̃4
̃ 0)
We noticed that 𝐽(𝜙,
=
(𝑎1 /4) ∑𝑛∈Z 𝜙
inf (𝜙,𝜓)∈𝑁𝐽(𝜙, 𝜓) and
∗
∗
1, 𝑛 = 𝑘,
={
0, 𝑛 ≠ 𝑘.
∗
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 2 .
≥ (𝜆 − 𝜔1 ) 󵄩󵄩󵄩󵄩𝜙
󵄩𝑙
(41)
for any (𝜙, 𝜓) ∈ 𝐸1 × 𝐸2 . Take (𝜙, 𝜓) = (𝑒(𝑘) , 0) and (𝜙, 𝜓) =
(0, 𝑒(𝑘) ) in (41) for 𝑘 ∈ Z, where
𝑒𝑛(𝑘)
(42)
𝑑𝑢𝑛
󵄨 󵄨2
󵄨 󵄨2
= −(A𝑢)𝑛 + 𝑏1𝑛 𝑢𝑛 − 𝑎1 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 𝑢𝑛 − 𝑎3 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 𝑢𝑛 ,
𝑑𝑡
𝑑V
󵄨 󵄨2
󵄨 󵄨2
𝑖 𝑛 = −(AV)𝑛 + 𝑏1𝑛 V𝑛 − 𝑎2 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 V𝑛 − 𝑎3 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 V𝑛 .
𝑑𝑡
(45)
󵄩 ̃ 󵄩󵄩2
󵄩 ̃ 󵄩󵄩2
󵄩
󵄩󵄩 − 𝜔2 󵄩󵄩󵄩𝜙
𝐷 = 󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2 .
(46)
󵄩 ̃ 󵄩󵄩2
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 − 𝜔2 󵄩󵄩󵄩𝜙
󵄩
𝐷 = 󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2
󵄩 ̃ 󵄩󵄩2
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 ̃ 󵄩󵄩2
󵄩
󵄩󵄩 − 𝜔1 󵄩󵄩󵄩𝜙
󵄩 󵄩
= 󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2 + (𝜔1 − 𝜔2 ) 󵄩󵄩𝜙󵄩󵄩𝑙2
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 2 ≤ 𝑎1 𝐵.
= 𝑎1 𝐵 + (𝜔1 − 𝜔2 ) 󵄩󵄩󵄩󵄩𝜙
󵄩𝑙
(47)
4
̃ ,
𝐵 = ∑𝜙
𝑛
𝑛∈Z
If 𝜔1 ≤ 𝜔2 < 𝜆, then
∗
By Theorem 4, the system (1) has a nontrivial solution.
However, it is uncertain if two components of this solution are
nonzero. Therefore, we want to find solutions of the system
(1) which have both of the components not identically zero.
In order to achieve this goal, we consider the system (1) with
𝑏1𝑛 = 𝑏2𝑛 , 𝑛 ∈ Z; that is,
=
For the sake of simplicity, we let
We see that 𝐽 (𝜙 , 𝜓 ) = 0. Thus, (𝜙 , 𝜓 ) is a nontrivial
solution to system (13). The proof is completed.
𝑖
(44)
∗
󵄩󵄩 ̃ 󵄩󵄩2
̃ 󵄩󵄩󵄩2
̃ 0) = 𝐻1 (𝜙,
̃ 0) = 󵄩󵄩󵄩󵄩𝜙
󵄩 󵄩
𝐻2 (𝜙,
󵄩 󵄩󵄩𝐸 − 𝜔1 󵄩󵄩𝜙󵄩󵄩𝑙2
Thus, Λ = 0 and
(𝐽󸀠 (𝜙∗ , 𝜓∗ ) , (𝜙, 𝜓)) = 0,
=
𝑛∈Z
(40)
󸀠
0, we have (𝑡∗ )2
󵄩 󵄩2 󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
𝐻1 (𝜙, 𝜓) = 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐸 − 𝜔1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝑙2 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐸 − 𝜔2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑙2 ,
= 2 ((𝐿 1 − 𝜔1 ) 𝜙∗ , 𝜙∗ ) + 2 ((𝐿 2 − 𝜔2 ) 𝜓∗ , 𝜓∗ )
4
(𝑎1 (𝜙𝑛∗ )
=
Thus, 𝑎3 > 𝑎1 and (47) yields 𝑎1 𝐵𝐷 < 𝑎1 𝑎3 𝐵2 .
If 𝜔2 < 𝜔1 < 𝜆, then by (45),
󵄩 ̃ 󵄩󵄩2
󵄩 ̃ 󵄩󵄩2
󵄩
󵄩󵄩 − 𝜔2 󵄩󵄩󵄩𝜙
𝐷 = 󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2
󵄩 ̃ 󵄩󵄩2
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 ̃ 󵄩󵄩2
󵄩
󵄩󵄩 − 𝜔1 󵄩󵄩󵄩𝜙
󵄩 󵄩
= 󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2 + (𝜔1 − 𝜔2 ) 󵄩󵄩𝜙󵄩󵄩𝑙2
(43)
In system (43), we know that 𝐿 1 = 𝐿 2 , where 𝐿 𝑖 , 𝑖 = 1, 2,
is given by (11). By the definition of 𝐸𝑖 , 𝑖 = 1, 2, in Section 2
of this paper, we obtain that 𝐸1 = 𝐸2 . Hence, 𝜆 1 = 𝜆 2 . For
the sake of simplicity, we let 𝐿 1 = 𝐿 2 = 𝐿, 𝐸1 = 𝐸2 = 𝐸,
and 𝜆 1 = 𝜆 2 = 𝜆. The notations in Section 2, such as
𝐽(𝜙, 𝜓), 𝐼(𝜙, 𝜓), and 𝑁 are the same.
Now, we give the second result of this paper as follows.
Theorem 5. Assume that 𝜔1 < 𝜆, 𝜔2 < 𝜆, 𝑎3 >
max {𝑎1 , 𝑎2 , ((𝜆 − 𝜔2 )/(𝜆 − 𝜔1 ))𝑎1 , ((𝜆 − 𝜔1 )/(𝜆 − 𝜔2 ))𝑎2 }, and
(10) holds. Then system (43) has a nontrivial standing wave
̃ 𝜓
̃ ≠ 0 and 𝜓
̃ ) in 𝐸 × 𝐸 with 𝜙
̃ ≠ 0.
solution (𝜙,
Proof. By Theorem 4, we know that system (43) has a noñ 𝜓
̃ ) in 𝐸 × 𝐸.
trivial standing wave solution (𝜙,
̃
̃ ≠ 0.
Now we will prove that 𝜙 ≠ 0 and 𝜓
̃ 𝜓
̃ 𝜓
̃ ) ∈ 𝑁, we know that (𝜙,
̃ ) ≠ (0, 0). If one of
Since (𝜙,
̃ 𝜓
̃ ≠ 0. For 𝜖 small
̃ ), say 𝜓
̃ = 0, then 𝜙
the components (𝜙,
̃
̃
enough, we consider (𝜙, 𝜖𝜙) ∈ (𝐸−{0})×(𝐸−{0}); by a similar
argument to the proof of Lemma 2, we know that there exists
̃ 𝑡∗ 𝜖𝜙)
̃ 𝑡∗ 𝜖𝜙)
̃ = 0; that is, (𝑡∗ 𝜙,
̃ ∈ 𝑁.
𝑡∗ such that 𝐼(𝑡∗ 𝜙,
(48)
󵄩 ̃ 󵄩󵄩2
󵄩󵄩 2
= 𝑎1 𝐵 + (𝜔1 − 𝜔2 ) 󵄩󵄩󵄩󵄩𝜙
󵄩𝑙
≤ 𝑎1 𝐵 +
𝜔1 − 𝜔2
𝜆 − 𝜔2
𝑎1 𝐵 =
𝑎 𝐵.
𝜆 − 𝜔1
𝜆 − 𝜔1 1
Thus, 𝑎3 > ((𝜆 − 𝜔2 )/(𝜆 − 𝜔1 ))𝑎1 and (48) yields 𝑎1 𝐵𝐷 <
𝑎1 𝑎3 𝐵2 .
From the above arguments, if 𝑎3 > max {𝑎1 , ((𝜆 − 𝜔2 )/(𝜆 −
𝜔1 ))𝑎1 }, then 𝑎1 𝐵𝐷 < 𝑎1 𝑎3 𝐵2 .
For 𝜖 small enough, we have
̃ 𝜖𝜙)
̃
𝐻12 (𝜙,
󵄩 ̃ 󵄩󵄩2 󵄩󵄩 ̃ 󵄩󵄩2
󵄩󵄩 ̃ 󵄩󵄩2 2
󵄩 ̃ 󵄩󵄩2
󵄩
󵄩󵄩 − 𝜔1 󵄩󵄩󵄩𝜙
󵄩 󵄩
󵄩 󵄩
= (󵄩󵄩󵄩󵄩𝜙
󵄩𝐸
󵄩 󵄩󵄩𝑙2 + 󵄩󵄩𝜖𝜙󵄩󵄩𝐸 − 𝜔2 󵄩󵄩𝜖𝜙󵄩󵄩𝑙2 )
2
= (𝑎1 𝐵 + 𝜖2 𝐷) = 𝑎12 𝐵2 + 2𝑎1 𝐵𝐷𝜖2 + 𝐷2 𝜖4
<
𝑎12 𝐵2
2 2
2 4
+ 2𝑎1 𝑎3 𝐵 𝜖 + 𝑎1 𝑎2 𝐵 𝜖
= 𝑎1 𝐵 (𝑎1 𝐵 + 𝑎2 𝐵𝜖4 + 2𝑎3 𝐵𝜖2 )
̃ 0) 𝐻2 (𝜙,
̃ 𝜖𝜙)
̃ .
= 𝐻2 (𝜙,
(49)
6
Abstract and Applied Analysis
Hence, by (49), we have
∗̃
̃ =
𝐽 (𝑡 𝜙, 𝑡 𝜖𝜙)
∗
̃ 𝜖𝜙)
̃
𝐻12 (𝜙,
1
̃ 0)
< 𝐻2 (𝜙,
̃
̃
4𝐻2 (𝜙, 𝜖𝜙) 4
[12] L. Liu, W. Yan, and X. Zhao, “The existence of standing wave
for the discrete coupled nonlinear Schrödinger lattice,” Physics
Letters A, vol. 374, no. 15-16, pp. 1690–1693, 2010.
(50)
̃ 0) = inf 𝐽 (𝜙, 𝜓) .
= 𝐽 (𝜙,
(𝜙,𝜓)∈𝑁
̃ ≠ 0.
This is a contradiction. So, 𝜓
̃ ≠ 0 and 𝑎3 > max {𝑎2 , ((𝜆−𝜔1 )/(𝜆−𝜔2 ))𝑎2 },
Similarly, if 𝜓
̃ ≠ 0. The proof is completed.
then 𝜙
Acknowledgments
This work is supported by Program for Changjiang Scholars
and Innovative Research Team in University (no. IRT1226),
the National Natural Science Foundation of China (no.
11171078), and the Specialized Fund for the Doctoral Program
of Higher Education of China (no. 20114410110002).
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