On the Existence of Ground State Solutions of the Periodic Discrete Coupled Nonlinear Schrödinger Lattice

Abstract

We study the existence of ground state solutions of the periodic discrete coupled nonlinear Schrödinger lattice by using the Nehari manifold approach combined with periodic approximations. We show that both of the components of the ground state solutions are not zero.

1. Introduction

In this paper, we consider the coupled discrete Schrödinger system 𝑖𝑑𝑢_𝑛()𝑑𝑡=−𝒜𝑢_𝑛+𝑏_1𝑛𝑢_𝑛−𝑎₁𝑢_𝑛²𝑢_𝑛−𝑎₃𝑣_𝑛²𝑢_𝑛,𝑖𝑑𝑣_𝑛()𝑑𝑡=−𝒜𝑣_𝑛+𝑏_2𝑛𝑣_𝑛−𝑎₂𝑣_𝑛²𝑣_𝑛−𝑎₃𝑢_𝑛²𝑣_𝑛, (1) where 𝑎_𝑖 is a positive constant, {𝑏_𝑗𝑛} is a real valued 𝑇-periodic sequence, 𝑖=1,2,3, and 𝑗=1,2. 𝒜 is the discrete Laplacian operator defined as (𝒜𝑢)_𝑛=𝑢_𝑛+1+𝑢_𝑛−1−2𝑢_𝑛.

System (1) could be viewed as the discretization of the two-component system of time-dependent nonlinear Gross-Pitaevskii system (see [1] for more detail) as 𝑖ℏ𝜕_𝑡ℏ𝑢=−²2𝑚Δ𝑢+𝑏₁(𝑥)𝑢−𝑎₁|𝑢|²𝑢−𝑎₃|𝑣|²𝑢,𝑖ℏ𝜕_𝑡ℏ𝑣=−²2𝑚Δ𝑣+𝑏₂(𝑥)𝑣−𝑎₂|𝑣|²𝑣−𝑎₃|𝑢|²𝑣. (2)

It is well known that coupled nonlinear Schrödinger equations arise quite naturally in nonlinear optics [2] and Bose-Einstein condensates. Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was realized in 1997 (see [3]), which stimulated various analytical and numerical results on the ground state solutions of 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 condensed-matter physics to biology. During the last years, there has been a growing interest in approaches to the existence problem for ground states. We refer to the continuation methods in [4, 5], which have been proved to be powerful for both theoretical considerations and numerical computations (see [6]), to [7], which exploits spatial dynamics and centre manifold reduction, to the variational methods in [8–15], which rely on critical point techniques (linking theorems, the Nehari manifold), and to the Krasnoselskii fixed point theorem together with a suitable compactness criterion [16].

The aim of this paper is to study the discrete solitons of (1), that is, solutions of the form 𝑢_𝑛=exp−𝑖𝜔₁𝑡𝜙_𝑛,𝑣_𝑛=exp−𝑖𝜔₂𝑡𝜓_𝑛,𝑛∈ℤ,lim_|𝑛|→∞𝑢_𝑛=0,lim_|𝑛|→∞𝑣_𝑛=0, (3)(4) where the amplitudes 𝜙_𝑛 and 𝜓_𝑛 are supposed to be real. Inserting the ansatz of the discrete solitons (3) into (1), we obtain the following equivalent algebraic equations: −𝒜𝜙_𝑛−𝜔₁𝜙_𝑛+𝑏_1𝑛𝜙_𝑛−𝑎₁𝜙_𝑛²𝜙_𝑛−𝑎₃𝜓_𝑛²𝜙_𝑛−=0,𝒜𝜓_𝑛−𝜔₂𝜓_𝑛+𝑏_2𝑛𝜓_𝑛−𝑎₂𝜓_𝑛²𝜓_𝑛−𝑎₃𝜙_𝑛²𝜓_𝑛=0, (5) and (4) becomes lim_|𝑛|→∞𝜙_𝑛=0,lim_|𝑛|→∞𝜓_𝑛=0. (6)

In fact, we consider the following more general equations: 𝐿₁𝜙_𝑛−𝜔₁𝜙_𝑛−𝑎₁𝜙_𝑛²𝜙_𝑛−𝑎₃𝜓_𝑛²𝜙_𝑛𝐿=0,₂𝜓_𝑛−𝜔₂𝜓_𝑛−𝑎₂𝜓_𝑛²𝜓_𝑛−𝑎₃𝜙_𝑛²𝜓_𝑛=0, (7) where 𝐿₁ and 𝐿₂ are the second-order difference operators defined by 𝐿₁𝜙_𝑛=𝛼_𝑛𝜙_𝑛+1+𝛼_𝑛−1𝜙_𝑛−1+𝛽_𝑛𝜙_𝑛,𝐿₂𝜓_𝑛=𝛾_𝑛𝜓_𝑛+1+𝛾_𝑛−1𝜓_𝑛−1+𝛿_𝑛𝜓_𝑛, (8) where {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are real valued 𝑇-periodic sequences, and 𝑎₁, 𝑎₂, and 𝑎₃ are positive numbers. Obviously, (5) is a special case of (7) with 𝛼_𝑛≡−1, 𝛽_𝑛≡2+𝑏_1𝑛, 𝛾_𝑛≡−1, and 𝛿_𝑛≡2+𝑏_2𝑛.

Since, for 𝑖=1,2, the operator 𝐿_𝑖 is a bounded and self-adjoint operator in 𝑙², its spectrum 𝜎(𝐿_𝑖) has a band structure; that is, 𝜎(𝐿_𝑖) is a union of a finite number of closed intervals [17]. The complement ℝ−𝜎(𝐿_𝑖) consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite which are denoted by (−∞,𝜃_1𝑖) and (𝜃_2𝑖,∞), respectively.

In this paper, we consider two types of solutions to (7) as follows: (i) 2𝑘𝑇-periodic, that is, 𝜙_{𝑛+2𝑘𝑇}=𝜙_𝑛, 𝜓_{𝑛+2𝑘𝑇}=𝜓_𝑛, and (ii) discrete solitons. Actually, in case (ii), we look for solutions in the space 𝑙²×𝑙²=𝑙²(ℤ)×𝑙²(ℤ); then (6) holds naturally. System (7) has a trivial solution 𝜙_𝑛≡0, 𝜓_𝑛≡0. We are looking for nontrivial solutions.

The main idea in this paper is as follows. First, we consider (7) in a finite 2𝑘𝑇-periodic sequence space, and 𝜔_𝑖 is not a spectrum of the corresponding operator 𝐿_𝑖, 𝑖=1,2. By using the Nehari manifold approach, we obtain the existence of 2𝑘𝑇-periodic solutions. Then we show that these solutions have upper and lower bounds. Finally, by an approximation technique, we prove that the limit of these solutions exists and is the solution of (7) in 𝑙²×𝑙². Compared with the existence of ground state solutions of the DNLS, the difficulty is that we need to show that both of the components of the ground state solutions are not zero.

The remaining of this paper is organized as follows. First, in Section 2, we establish the variational framework associated with (7) and introduce the Nehari manifolds. Then, in Section 3, we present a sufficient condition on the existence of 2𝑘𝑇-periodic solutions and nontrivial solutions in 𝑙²×𝑙² of (7).

2. Preliminaries and the Nehari Manifold

In this section, we first establish the variational framework associated with (7).

Let 𝑆 be the set of the following form: 𝑆=𝜙=𝜙_𝑛∶𝜙_𝑛∈ℝ,𝑛∈ℤ. (9)

For any fixed positive integer 𝑘, we define the subspace 𝑋_𝑘 of 𝑆 as 𝑋_𝑘=𝜙∈𝑆∶𝜙_{𝑛+2𝑘𝑇}=𝜙_𝑛,𝑛∈ℤ. (10) Obviously, 𝑋_𝑘 is isomorphic to 𝑅^2𝑘𝑇 and hence 𝑋_𝑘 can be equipped with the inner product (⋅,⋅)_𝑘 and norm ‖⋅‖_𝑘 as 𝜙,𝜓_𝑘=^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝜙_𝑛𝜓_𝑛,𝜙,𝜓∈𝑋_𝑘,𝜙_𝑘=^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝜙²_𝑛^1/2,𝜙∈𝑋_𝑘, (11)(12) respectively. Sometimes, we will consider 𝑙^𝑝 norm on 𝑋_𝑘 as 𝜙_𝑘𝑝=^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝜙_𝑛^𝑝1/𝑝,1≤𝑝<∞. (13) We also define a norm ‖⋅‖_𝑘∞ on 𝑋_𝑘 by 𝜙_𝑘∞=max𝜙_𝑛∶−𝑘𝑇≤𝑛≤𝑘𝑇−1,𝜙∈𝑋_𝑘. (14) The symbol 𝑋 stands for the space 𝑋=𝑙² with the norm 𝜙=_𝑛∈ℤ𝜙²_𝑛^1/2,𝜙∈𝑋, (15) and the inner product 𝜙,𝜓=_𝑛∈ℤ𝜙_𝑛𝜓_𝑛,𝜙,𝜓∈𝑋. (16) We also consider 𝑙^𝑝 norm on 𝑋 as 𝜙_𝑝=_𝑛∈ℤ𝜙_𝑛^𝑝1/𝑝𝜙,1≤𝑝<∞,_∞=max𝜙_𝑛∶𝑛∈ℤ. (17) We mention that 𝜙_𝑘𝑝≤𝜙_𝑘𝑞,𝜙_𝑝≤𝜙_𝑞, (18) where 1≤𝑞≤𝑝≤∞.

Consider the functionals 𝐽_𝑘 on 𝑋_𝑘×𝑋_𝑘 and 𝐽 on 𝑋×𝑋 defined by 𝐽_𝑘=1𝜙,𝜓2𝐿₁−𝜔₁𝜙,𝜙_𝑘+12𝐿₂−𝜔₂𝜓,𝜓_𝑘−14^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛,𝐽=1𝜙,𝜓2𝐿₁−𝜔₁+1𝜙,𝜙2𝐿₂−𝜔₂−1𝜓,𝜓4_𝑛∈ℤ𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛, (19)(20) respectively. Then 𝐽_𝑘(𝜙,𝜓)∈𝐶¹(𝑋_𝑘×𝑋_𝑘,ℝ) and its derivative is given by 𝐽^′_𝑘𝜙,𝜓,𝜙,𝜓=𝐿₁−𝜔₁𝜙,𝜙_𝑘+𝐿₂−𝜔₂𝜓,𝜓_𝑘−^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛, (21) and 𝐽(𝜙,𝜓)∈𝐶¹(𝑋×𝑋,ℝ) and its derivative is given by 𝐽^′𝜙,𝜓,𝜙,𝜓=𝐿₁−𝜔₁𝜙,𝜙+𝐿₂−𝜔₂𝜓,𝜓−_𝑛∈ℤ𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛. (22)

Let 𝜁_𝑖 be the distance from 𝜔_𝑖 to the spectrum 𝜎(𝐿_𝑖); that is, 𝜁_𝑖=𝜃_1𝑖−𝜔_𝑖,𝑖=1,2. (23) Let 𝜁=min{𝜁₁,𝜁₂}. Then, we have 𝐿_𝑖−𝜔_𝑖𝜙,𝜙_𝑘≥𝜁𝜙²_𝑘,𝜙∈𝑋_𝑘𝐿,𝑖=1,2,_𝑖−𝜔_𝑖𝜙,𝜙≥𝜁𝜙²,𝜙∈𝑋,𝑖=1,2. (24)(25) Furthermore, we let 𝜆_1∗=min_𝑛∈ℤ𝛽_𝑛−𝜔₁−𝛼_𝑛−1−𝛼_𝑛,𝜆^∗₁=max_𝑛∈ℤ𝛽_𝑛−𝜔₁+𝛼_𝑛−1+𝛼_𝑛,𝜆_2∗=min_𝑛∈ℤ𝛿_𝑛−𝜔₂−𝛾_𝑛−1−𝛾_𝑛,𝜆^∗₂=max_𝑛∈ℤ𝛿_𝑛−𝜔₂+𝛾_𝑛−1+𝛾_𝑛. (26) Then, obviously, 𝜆^∗₁>0 and 𝜆^∗₂>0.

Denote 𝜆=max_𝑛∈ℤ𝛼_𝑛−𝛾_𝑛+𝛼_𝑛−1−𝛾_𝑛−1+𝛽_𝑛−𝛿_𝑛. (27) Then 𝐿₁−𝐿₂𝜙≤𝜆𝜙,𝜙∈𝑋. (28)

Next, we study the main properties of the Nehari manifolds with the functionals 𝐽_𝑘 and 𝐽.

Let 𝐼_𝑘𝜙,𝜓=𝐽^′_𝑘𝜙,𝜓,𝜙,𝜓,𝜙,𝜓∈𝑋_𝑘×𝑋_𝑘,𝐼𝜙,𝜓=𝐽^′𝜙,𝜓,𝜙,𝜓,𝜙,𝜓∈𝑋×𝑋. (29) Then 𝐼_𝑘 and 𝐼 are 𝐶¹ functionals and their derivatives are given by 𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓=2𝐿₁−𝜔₁𝜙,𝜙_𝑘+2𝐿₂−𝜔₂𝜓,𝜓_𝑘−4^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛,𝐼^′𝜙,𝜓,𝜙,𝜓=2𝐿₁−𝜔₁𝜙,𝜙+2𝐿₂−𝜔₂𝜓,𝜓−4_𝑛∈ℤ𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛, (30)(31) respectively. The Nehari manifolds are defined as follows: 𝑁_𝑘=𝜙,𝜓∈𝑋_𝑘×𝑋_𝑘∶𝐼_𝑘,.𝜙,𝜓=0,𝜙,𝜓≠0𝑁=𝜙,𝜓∈𝑋×𝑋∶𝐼𝜙,𝜓=0,𝜙,𝜓≠0 (32) Note that 𝑁 contains all critical points of 𝐽 in 𝑋×𝑋.

To prove the main results, we need some lemmas on the Nehari manifolds.

Lemma 1. Assume that 𝜔₁<𝜃₁₁ and 𝜔₂<𝜃₁₂ hold. Then the sets 𝑁_𝑘 and 𝑁 are nonempty closed 𝐶¹ submanifolds in 𝑋_𝑘×𝑋_𝑘 and 𝑋×𝑋, respectively. The derivatives 𝐼^′_𝑘 and 𝐼^′ are nonzero on the corresponding Nehari manifolds. Moreover, there exists 𝜂>0 such that ‖𝜙‖²_𝑘+‖𝜓‖²_𝑘≥𝜂, (𝜙,𝜓)∈𝑁_𝑘 and ‖𝜙‖²+‖𝜓‖²≥𝜂, (𝜙,𝜓)∈𝑁.

Proof. The proofs for both cases are similar. We only provide the proof for the case of 𝑁_𝑘 as an illustration.
First, we show that 𝑁_𝑘≠∅.
Let (𝜙,𝜓)∈𝑋_𝑘×𝑋_𝑘−{(0,0)}. Then by (21) 𝐼_𝑘𝑡𝜙,𝑡𝜓=𝑡²𝐿₁−𝜔₁𝜙,𝜙_𝑘+𝐿₂−𝜔₂𝜓,𝜓_𝑘−𝑡^{2𝑘𝑇−1}_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛. (33) Noticing that ((𝐿₁−𝜔₁)𝜙,𝜙)_𝑘>0 and ((𝐿₂−𝜔₂)𝜓,𝜓)_𝑘>0, by (33), we see that 𝐼_𝑘(𝑡𝜙,𝑡𝜓)>0 for 𝑡>0 small enough and 𝐼_𝑘(𝑡𝜙,𝑡𝜓)<0 for 𝑡>0 large enough. As a consequence, there exists 𝑡₀>0 such that 𝐼_𝑘(𝑡₀𝜙,𝑡₀𝜓)=0; that is, (𝑡₀𝜙,𝑡₀𝜓)∈𝑁_𝑘.
Let (𝜙,𝜓)∈𝑁_𝑘. By (21), (30), and the definition of 𝑁_𝑘, we have 𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓=𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓−2𝐼_𝑘𝜙,𝜓=−2^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛<0. (34) Hence, 𝐼^′_𝑘≠0, and the implicit function theorem implies that 𝑁_𝑘 is a 𝐶¹ submonifold in 𝑋_𝑘×𝑋_𝑘.
Now let us prove the last statement of the lemma. Let (𝜙,𝜓)∈𝑁_𝑘. By the assumption of Lemma 1 and the definition of 𝑁_𝑘, we have 𝐿₁−𝜔₁𝜙,𝜙_𝑘+𝐿₂−𝜔₂𝜓,𝜓_𝑘=^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛≤𝑎^∗𝜙²_𝑘4+𝜓²_𝑘4²≤𝑎^∗𝜙²_𝑘+𝜓²_𝑘², (35) where 𝑎^∗=max{𝑎₁,𝑎₂,𝑎₃}. Let 𝜂=𝜁/𝑎^∗>0. Then, by (24) and (35), it is easy to see that 𝜁𝜙²_𝑘+𝜓²_𝑘≤𝑎^∗𝜙²_𝑘+𝜓²_𝑘², (36) which implies that 𝜙²_𝑘+𝜓²_𝑘≥𝜁𝑎^∗=𝜂. (37)
Closedness of 𝑁_𝑘 is obvious. The proof is completed.

Lemma 2. Assume that 𝜔₁<𝜃₁₁ and 𝜔₂<𝜃₁₂ hold. Then there exists 𝜌>0 such that 𝐽_𝑘(𝜙,𝜓)≥𝜌 for all (𝜙,𝜓)∈𝑁_𝑘.

Proof. For any (𝜙,𝜓)∈𝑁_𝑘, we have 𝐽_𝑘𝜙,𝜓=𝐽_𝑘−1𝜙,𝜓4𝐼_𝑘=1𝜙,𝜓4𝐿₁−𝜔₁𝜙,𝜙_𝑘+𝐿₂−𝜔₂𝜓,𝜓_𝑘≥𝜁4𝜙²_𝑘+𝜓²_𝑘. (38) By Lemma 1, we know that ‖𝜙‖²_𝑘+‖𝜓‖²_𝑘≥𝜂. Hence, let 𝜌=(𝜁𝜂)/4>0. Then (38) implies that 𝐽_𝑘(𝜙,𝜓)≥𝜌. The proof is completed.

Lemma 3. For (𝜙,𝜓)∈𝑁_𝑘, the function 𝐽_𝑘(𝑡𝜙,𝑡𝜓), 𝑡>0, has a unique critical point at 𝑡=1, which is, actually, a global maximum. The same statement holds for 𝑁 and 𝐽.

Proof. Let 𝐹(𝑡)=𝐽_𝑘(𝑡𝜙,𝑡𝜓), 𝑡>0, (𝜙,𝜓)∈𝑁_𝑘. Computing the derivative of 𝐹, we have 𝐹^′(𝑡)=𝑡1−𝑡^{2𝑘𝑇−1}_{𝑛=−𝑘𝑇}𝑎₁𝜙⁴_𝑛+𝑎₂𝜓⁴_𝑛+2𝑎₃𝜙²_𝑛𝜓²_𝑛. (39) This shows that 𝑡=1 is a unique maximum point. The proof is completed.

Lemma 4. Let (𝜙,𝜓) be a minimizer of the functional 𝐽_𝑘(𝜙,𝜓) constrained on the Nehari manifold 𝑁_𝑘; that is, 𝐽_𝑘𝜙,𝜓=𝑚_𝑘=inf_{(𝜙,𝜓)∈𝑁}𝐽_𝑘𝜙,𝜓, (40) and then (𝜙,𝜓) is a nontrivial 2𝑘𝑇-periodic solution to (7), which is called a nontrivial periodic ground state solution to (7).

Proof. According to Lagrange multiplier method, (𝜙,𝜓) is the critical point of the functional 𝐽_𝑘(𝜙,𝜓)+Λ𝐼_𝑘(𝜙,𝜓). Thus 𝐼_𝑘(𝜙,𝜓)=0, and for arbitrary (𝜙,𝜓)∈𝑋_𝑘×𝑋_𝑘, 𝐽^′_𝑘𝜙,𝜓+Λ𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓=0. (41) After taking (𝜙,𝜓)=(𝜙,𝜓), we obtain Λ𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓=0. (42) But 𝐼^′_𝑘𝜙,𝜓,𝜙,𝜓=2𝐿₁−𝜔₁𝜙,𝜙_𝑘+2𝐿₂−𝜔₂𝜓,𝜓_𝑘−4^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛²=−2^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛²<0. (43) Thus, Λ=0 and 𝐽^′_𝑘𝜙,𝜓,𝜙,𝜓=0, (44) for any (𝜙,𝜓)∈𝑋_𝑘×𝑋_𝑘. Take (𝜙,𝜓)=(𝑒^(𝑗),0) and (𝜙,𝜓)=(0,𝑒^(𝑗)) in (44) for 𝑗∈ℤ∩[−𝑘𝑇,𝑘𝑇−1], where 𝑒_𝑛^(𝑗)=1,𝑛=𝑗,0,𝑛≠𝑗. (45) We see that 𝐽^′_𝑘(𝜙,𝜓)=0. Thus, (𝜙,𝜓) is a nontrivial 2𝑘𝑇-periodic ground state solution to (7). The proof is completed.

Through a similar argument to the proof of Lemma 4, we get the following lemma.

Lemma 5. Let (𝜙,𝜓) be a minimizer of the functional 𝐽(𝜙,𝜓) constrained on the Nehari manifold 𝑁; that is, 𝐽𝜙,𝜓=𝑚=inf_{(𝜙,𝜓)∈𝑁}𝐽_𝑘𝜙,𝜓, (46) and then (𝜙,𝜓) is a nontrivial discrete soliton to (7), which is called a nontrivial ground state solution to (7).

3. Main Results

In this section, we will establish some sufficient conditions on the existence of 2𝑘𝑇-periodic solutions and nontrivial solutions in 𝑋×𝑋 of (7).

We start with the following.

Lemma 6. Assume that 𝜔₁<𝜃₁₁ and 𝜔₂<𝜃₁₂ hold. Then the minimum value in (40) is attained.

Proof. Let {(𝜙^(𝑗),𝜓^(𝑗))}⊂𝑁_𝑘 be a minimizing sequence for 𝐽_𝑘; that is, 𝐽_𝑘(𝜙^(𝑗),𝜓^(𝑗))→𝑚_𝑘 as 𝑗→+∞. The fact that 𝐽_𝑘𝜙^(𝑗),𝜓^(𝑗)=𝐽_𝑘𝜙^(𝑗),𝜓^(𝑗)−12𝐼_𝑘𝜙^(𝑗),𝜓^(𝑗)=14^𝑘𝑇−1_{𝑛=−𝑘𝑇}𝑎₁𝜙_𝑛^(𝑗)4+𝑎₂𝜓_𝑛^(𝑗)4+2𝑎₃𝜙_𝑛^(𝑗)2𝜓_𝑛^(𝑗)2 (47) shows that ‖(𝜙^(𝑗),𝜓^(𝑗))‖_𝑘∞ is bounded. Since the space 𝑋_𝑘×𝑋_𝑘 is finite dimensional, so the norm ‖(⋅,⋅)‖_𝑘∞ is equivalent to the Euclidean norm on 𝑋_𝑘×𝑋_𝑘, and the sequence {(𝜙^(𝑗),𝜓^(𝑗))} is bounded. Passing to a subsequence, we can assume that (𝜙^(𝑗),𝜓^(𝑗)) converges to (𝜙,𝜓)∈𝑋_𝑘×𝑋_𝑘. Since the set 𝑁_𝑘 is closed and the functional 𝐽_𝑘 is continuous, we obtain that (𝜙,𝜓)∈𝑁_𝑘 and 𝐽_𝑘(𝜙,𝜓)=𝑚_𝑘. The proof is completed.

To obtain a nontrivial solutions in 𝑋×𝑋 of (7), we need the following lemma.

Lemma 7. Assume that 𝜔₁<𝜃₁₁ and 𝜔₂<𝜃₁₂ hold. Let (𝜙^(𝑘),𝜓^(𝑘)) be a 2𝑘𝑇-periodic ground state solution, that is, a solution of (40). Then the sequences {𝑚_𝑘} and {‖𝜙^(𝑘)‖²_𝑘+‖𝜓^(𝑘)‖²_𝑘} are bounded above and below away from zero.

Proof. By Lemma 2, {𝑚_𝑘} is obviously bounded below away from zero. Let 𝜙_𝑛^(𝑘),𝜓_𝑛^(𝑘)=()()[]−{0},1,0,𝑛=0,0,0,𝑛∈ℤ∩−𝑘𝑇,𝑘𝑇−1 (48) so (𝜙^(𝑘),𝜓^(𝑘))∈𝑋_𝑘×𝑋_𝑘−{(0,0)}, as in the beginning of the proof of Lemma 1; there exists 𝑡_𝑘=(𝛽₀−𝜔₁)/𝑎₁ such that 𝐼_𝑘(𝑡_𝑘𝜙^(𝑘),𝑡_𝑘𝜓^(𝑘))=0; that is, (𝑡_𝑘𝜙^(𝑘),𝑡_𝑘𝜓^(𝑘))∈𝑁_𝑘. Therefore, (𝛽₀−𝜔₁)²/(4𝑎₁)=𝐽_𝑘(𝑡_𝑘𝜙^(𝑘),𝑡_𝑘𝜓^(𝑘))≥𝑚_𝑘. And 𝑚_𝑘=𝐽_𝑘(𝜙^(𝑘),𝜓^(𝑘)) is bounded above.
Next, we prove that {‖𝜙^(𝑘)‖²_𝑘+‖𝜓^(𝑘)‖²_𝑘} are bounded above and below away from zero. By the previous proof, we see that 𝑚_𝑘=𝐽_𝑘(𝜙^(𝑘),𝜓^(𝑘)) is bounded above and below away from zero; that is, there exist 𝜌>0 and 𝜚>0 such that 𝜌≤𝐽_𝑘(𝜙^(𝑘),𝜓^(𝑘))≤𝜚. Then 𝐽_𝑘𝜙^(𝑘),𝜓^(𝑘)=𝐽_𝑘𝜙^(𝑘),𝜓^(𝑘)−14𝐼_𝑘𝜙^(𝑘),𝜓^(𝑘)=14𝐿₁−𝜔₁𝜙^(𝑘),𝜙^(𝑘)_𝑘+14𝐿₂−𝜔₂𝜓^(𝑘),𝜓^(𝑘)_𝑘≥𝜁4𝜙^(𝑘)2_𝑘+𝜓^(𝑘)2_𝑘. (49) This implies that 𝜙^(𝑘)2_𝑘+𝜓^(𝑘)2_𝑘≤4𝜚𝜁. (50) The proof is completed.

Lemma 8. Assume that 𝜆_1∗>0 and 𝜆_2∗>0 hold. Let (𝜙^(𝑘),𝜓^(𝑘)) be a 2𝑘𝑇-periodic ground state solution, that is, a solution of (40). Then there exist positive constants 𝜇 and 𝜈 such that 𝜇≤max𝜙^(𝑘)_𝑘∞,𝜓^(𝑘)_𝑘∞≤𝜈. (51)

Proof. By Lemmas 4 and 6, we know that (𝜙^(𝑘),𝜓^(𝑘)) is a nontrivial critical point of 𝐽_𝑘. Therefore, we have 𝐿₁𝜙^(𝑘)_𝑛−𝜔₁𝜙_𝑛^(𝑘)=𝑎₁𝜙_𝑛^(𝑘)2𝜙_𝑛^(𝑘)+𝑎₃𝜓_𝑛^(𝑘)2𝜙_𝑛^(𝑘),𝐿₂𝜓^(𝑘)_𝑛−𝜔₂𝜓_𝑛^(𝑘)=𝑎₂𝜓_𝑛^(𝑘)2𝜓_𝑛^(𝑘)+𝑎₃𝜙_𝑛^(𝑘)2𝜓_𝑛^(𝑘). (52) Let 𝑚_𝑖∈ℤ with −𝑘𝑇≤𝑚_𝑖≤𝑘𝑇−1, 𝑖=1,2 such that ‖𝜙^(𝑘)‖_𝑘∞=|𝜙_𝑚^(𝑘)₁| and ‖𝜓^(𝑘)‖_𝑘∞=|𝜓_𝑚^(𝑘)₂|. By the fact that 𝐿₁𝜙^(𝑘)_𝑚1−𝜔₁𝜙_𝑚^(𝑘)₁=𝑎₁𝜙_𝑚^(𝑘)₁²𝜙_𝑚^(𝑘)₁+𝑎₃𝜓_𝑚^(𝑘)₁²𝜙_𝑚^(𝑘)₁,𝐿₂𝜓^(𝑘)_𝑚2−𝜔₂𝜓_𝑚^(𝑘)₂=𝑎₂𝜓_𝑚^(𝑘)₂²𝜓_𝑚^(𝑘)₂+𝑎₃𝜙_𝑚^(𝑘)₂²𝜓_𝑚^(𝑘)₂, (53) we get 𝐿₁−𝜔₁𝜙_𝑚^(𝑘)₁=𝑎₁𝜙^(𝑘)3_𝑘∞+𝑎₃𝜓_𝑚^(𝑘)₁²𝜙^(𝑘)_𝑘∞,𝐿₂−𝜔₂𝜓_𝑚^(𝑘)₂=𝑎₂𝜓^(𝑘)3_𝑘∞+𝑎₃𝜙_𝑚^(𝑘)₂²𝜓^(𝑘)_𝑘∞. (54)(55) If one of the components of (𝜙^(𝑘),𝜓^(𝑘)), say 𝜓^(𝑘), is equal to 0, then 𝜙^(𝑘)≠0. Thus, by (54), we obtain 𝜆_1∗≤𝑎₁𝜙^(𝑘)2_𝑘∞,𝜆^∗₁≥𝑎₁𝜙^(𝑘)2_𝑘∞. (56) By (56), we get 𝜆_1∗𝑎₁≤𝜙^(𝑘)_𝑘∞≤𝜆^∗₁𝑎₁. (57) Similarly, if 𝜓^(𝑘)≠0, then we have 𝜆_2∗𝑎₂≤𝜓^(𝑘)_𝑘∞≤𝜆^∗₂𝑎₂. (58)
If 𝜙^(𝑘)≠0 and 𝜓^(𝑘)≠0, then, by (54) and (55), we obtain 𝜆_1∗≤𝑎₁𝜙^(𝑘)2_𝑘∞+𝑎₃𝜓^(𝑘)2_𝑘∞,𝜆^∗₁≥𝑎₁𝜙^(𝑘)2_𝑘∞,𝜆_2∗≤𝑎₃𝜙^(𝑘)2_𝑘∞+𝑎₂𝜓^(𝑘)2_𝑘∞,𝜆^∗₂≥𝑎₂𝜓^(𝑘)2_𝑘∞. (59)
By (59), we get max𝜆_1∗𝑎₁+𝑎₃,𝜆_2∗𝑎₂+𝑎₃≤max𝜙^(𝑘)_𝑘∞,𝜓^(𝑘)_𝑘∞≤max𝜆^∗₁𝑎₁,𝜆^∗₂𝑎₂. (60)
Let 𝜇=min𝜆_1∗𝑎₁,𝜆_2∗𝑎₂,max𝜆_1∗𝑎₁+𝑎₃,𝜆_2∗𝑎₂+𝑎₃,𝜈=max𝜆^∗₁𝑎₁,𝜆^∗₂𝑎₂. (61) Then, by (57), (58), and (60), we get (51). The proof is completed.

Now we are ready to state our main results.

Theorem 9. Assume that 𝜔₁<𝜃₁₁ and 𝜔₂<𝜃₁₂ hold. Then for every positive integer 𝑘, (7) possesses a nontrivial 2𝑘𝑇-periodic ground state solution (𝜙^(𝑘),𝜓^(𝑘)) in 𝑋_𝑘×𝑋_𝑘. Moreover, there are other three nontrivial 2𝑘𝑇-periodic ground state solutions (−𝜙^(𝑘),𝜓^(𝑘)), (𝜙^(𝑘),−𝜓^(𝑘)), and (−𝜙^(𝑘),−𝜓^(𝑘)) to (7).

Proof. The proof follows easily from Lemmas 4 and 6.

Theorem 10. Assume that 𝜔_𝑖<𝜃_1𝑖, 𝜆_𝑖∗>0, 𝑖=1,2, and 𝑎₃>max𝜆+𝜃₁₁−𝜔₁𝜃₁₁−𝜔₁𝑎₁,𝜆+𝜃₁₁−𝜔₂𝜃₁₁−𝜔₁𝑎₁,𝜆+𝜃₁₂−𝜔₁𝜃₁₂−𝜔₂𝑎₂,𝜆+𝜃₁₂−𝜔₂𝜃₁₂−𝜔₂𝑎₂ (62) holds. Then system (7) has a nontrivial ground state solution (𝜙^∗,𝜓^∗) in 𝑋×𝑋 with 𝜙^∗≠0 and 𝜓^∗≠0. Moreover, there are other three nontrivial ground state solutions (−𝜙^∗,𝜓^∗), and (𝜙^∗,−𝜓^∗), (−𝜙^∗,−𝜓^∗) to (7).

Proof. Consider the sequence {(𝜙^(𝑘),𝜓^(𝑘))} of 2𝑘𝑇-periodic solutions found in Lemma 6. By Lemma 8, without loss of generality, we can assume that the subsequence {(𝜙^(𝑘𝑖),𝜓^(𝑘𝑖))} of {(𝜙^(𝑘),𝜓^(𝑘))} satisfies 𝜇≤‖𝜙^(𝑘𝑖)‖_𝑘∞≤𝜈. Thus, there exists 𝑛_𝑘𝑖∈ℤ such that 𝜙^(𝑘𝑖)_𝑛𝑘𝑖≥𝜇. (63) By the periodicity of the coefficients in (7), we see that {(𝜙^(𝑘)_𝑛+𝑇,𝜓^(𝑘)_𝑛+𝑇)} is also a solution to (7). Making some shifts if necessary, without loss of generality, we can assume that 0≤𝑛_𝑘𝑖≤𝑇−1 in (63). Moreover, passing to a subsequence {𝜙^{(𝑘𝑖𝑗)}} of {𝜙^(𝑘𝑖)}, we can also assume that 𝑛_𝑘𝑖=𝑛^∗ and 0≤𝑛^∗≤𝑇−1. It follows from (37), (50), (51), and (63) that we can choose a subsequence, still denoted by {𝜙^(𝑘)} and {𝜓^(𝑘)}, such that 𝜙_𝑛^(𝑘)→𝜙^∗_𝑛 and 𝜓_𝑛^(𝑘)→𝜓^∗_𝑛 for all 𝑛∈ℤ. Notice that |𝜙^∗_𝑛∗|≥𝜇. Then (𝜙^∗,𝜓^∗)∈𝑋×𝑋 with (𝜙^∗,𝜓^∗)≠(0,0). Furthermore, (7) possesses pointwise limits and hence (𝜙^∗,𝜓^∗) is a nontrivial solution to (7). By the way, (−𝜙^∗,𝜓^∗), (𝜙^∗,−𝜓^∗), and (−𝜙^∗,−𝜓^∗) are nontrivial solutions to (7). Since 𝐽(𝜙^∗,𝜓^∗)=𝐽(−𝜙^∗,𝜓^∗)=𝐽(𝜙^∗,−𝜓^∗)=𝐽(−𝜙^∗,−𝜓^∗), if (𝜙^∗,𝜓^∗) is a ground state solution, then (−𝜙^∗,𝜓^∗), (𝜙^∗,−𝜓^∗), and (−𝜙^∗,−𝜓^∗) are the ground state solutions.
Now we prove that the solution (𝜙^∗,𝜓^∗) is a ground state solution. By Lemma 5, we have to show that 𝐽(𝜙^∗,𝜓^∗)=𝑚. Actually, we have proven that, for every sequence 𝑘_𝑗→∞, passing to a subsequence still denoted by 𝑘_𝑗 and making appropriate shifts, we can suppose that 𝜙^(𝑘𝑗)_𝑛→𝜙^∗_𝑛 and 𝜓^(𝑘𝑗)_𝑛→𝜓^∗_𝑛 pointwise, where (𝜙^∗,𝜓^∗)∈𝑋×𝑋 is a nontrivial solution. For any positive integer 𝐾, we have liminf_𝑗→∞𝐽_𝑘𝑗𝜙^(𝑘𝑗),𝜓^(𝑘𝑗)=liminf_𝑗→∞14^{𝑘𝑗𝑇−1}_{𝑛=−𝑘𝑗𝑇}𝑎₁𝜙^(𝑘𝑗)_𝑛⁴+𝑎₂𝜓^(𝑘𝑗)_𝑛⁴+2𝑎₃𝜙^(𝑘𝑗)_𝑛²𝜓^(𝑘𝑗)_𝑛²≥liminf_𝑗→∞14^𝐾𝑇−1_{𝑛=−𝐾𝑇}𝑎₁𝜙^(𝑘𝑗)_𝑛⁴+𝑎₂𝜓^(𝑘𝑗)_𝑛⁴+2𝑎₃𝜙^(𝑘𝑗)_𝑛²𝜓^(𝑘𝑗)_𝑛²=14^𝐾𝑇−1_{𝑛=−𝐾𝑇}𝑎₁𝜙^∗_𝑛⁴+𝑎₂𝜓^∗_𝑛⁴+2𝑎₃𝜙^∗_𝑛²𝜓^∗_𝑛². (64) Let 𝐾→∞; we obtain that liminf_𝑗→∞𝐽_𝑘𝑗𝜙^(𝑘𝑗),𝜓^(𝑘𝑗)≥𝐽𝜙^(∗),𝜓^(∗)≥𝑚, (65) and, hence, liminf_𝑘→∞𝑚_𝑘≥𝑚. (66) Now we prove that limsup_𝑘→∞𝑚_𝑘≤𝑚. (67) By Lemma 7, we know that the sequence {𝑚_𝑘} is bounded above and below away from zero. We extract a subsequence, still denoted by {𝑚_𝑘}, and hence we prove that lim_𝑘→∞𝑚_𝑘≤𝑚. (68) Given 𝜏>0, let (𝜙,𝜓)∈𝑁 be such that 𝐽𝜓=1𝜙,4_𝑛∈ℤ𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛²<𝑚+𝜏. (69) Choose 𝑡₁>1 sufficiently close to 1 such that 14_𝑛∈ℤ𝑎₁𝑡₁𝜙_𝑛⁴+𝑎₂𝑡₁𝜓_𝑛⁴+2𝑎₃𝑡₁𝜙_𝑛²𝑡₁𝜓_𝑛²<𝑚+𝜏. (70) We also have that 𝐼(𝑡₁𝜙,𝑡₁𝜓)<0. By density argument, we can find a finitely supported sequence (𝜙,𝜓) sufficiently close to (𝑡₁𝜙,𝑡₁𝜓) in 𝑋 such that 𝐼(𝜙,𝜓)<0 and 14_𝑛∈ℤ𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛²<𝑚+𝜏. (71) Then there exists 𝑡₂∈(0,1) such that 𝐼(𝑡₂𝜙,𝑡₂𝜓)=0 and 𝐽𝑡₂𝜙,𝑡₂𝜓=14_𝑛∈ℤ𝑎₁𝑡₂𝜙_𝑛⁴+𝑎₂𝑡₂𝜓_𝑛⁴+2𝑎₃𝑡₂𝜙_𝑛²𝑡₂𝜓_𝑛²<14_𝑛∈ℤ𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛²<𝑚+𝜏. (72) Let (𝜙^(𝑘),𝜓^(𝑘))∈𝑋_𝑘×𝑋_𝑘 be such that 𝜙_𝑛^(𝑘)=𝜙_𝑛 and 𝜓_𝑛^(𝑘)=𝜓_𝑛 if 𝑛∈ℤ∩[−𝑘𝑇,𝑘𝑇−1]. If 𝑘 is large enough, then 𝐼_𝑘(𝑡₂𝜙^(𝑘),𝑡₂𝜓^(𝑘))=𝐼(𝑡₂𝜙,𝑡₂𝜓)=0 and 𝐽_𝑘𝑡₂𝜙^(𝑘),𝑡₂𝜓^(𝑘)=𝐽𝑡₂𝜙,𝑡₂𝜓<𝑚+𝜏. (73) This implies (68). Hence, by (66) and (68), we have 𝐽(𝜙^∗,𝜓^∗)=𝑚.
Finally, we will show that 𝜙^∗≠0 and 𝜓^∗≠0. From the above arguments, system (7) has a nontrivial ground state solution (𝜙^∗,𝜓^∗) in 𝑋×𝑋. If one of the components of (𝜙^∗,𝜓^∗), say 𝜓^∗, is equal to 0, then 𝜙^∗≠0 (the proof for the other case is similar). For 𝜖 small enough, we consider (𝜙^∗,𝜖𝜙^∗)∈(𝑋−{0})×(𝑋−{0}). By a similar argument to the proof of Lemma 1, we know that there exists 𝑡^∗ such that 𝐼(𝑡^∗𝜙^∗,𝑡^∗𝜖𝜙^∗)=0; that is (𝑡^∗𝜙^∗,𝑡^∗𝜖𝜙^∗)∈𝑁. By 𝐼(𝑡^∗𝜙^∗,𝑡^∗𝜖𝜙^∗)=0, we have (𝑡^∗)²=𝐻₁(𝜙^∗,𝜖𝜙^∗)/𝐻₂(𝜙^∗,𝜖𝜙^∗), where 𝐻₁𝜙,𝜓=𝐿₁−𝜔₁𝜙,𝜙+𝐿₂−𝜔₂,𝐻𝜓,𝜓₂𝜙,𝜓=_𝑛∈ℤ𝑎₁𝜙_𝑛⁴+𝑎₂𝜓_𝑛⁴+2𝑎₃𝜙_𝑛²𝜓_𝑛². (74) Moreover, 𝐽(𝑡^∗𝜙^∗,𝑡^∗𝜖𝜙^∗)=𝐻²₁(𝜙^∗,𝜖𝜙^∗)/(4𝐻₂(𝜙^∗,𝜖𝜙^∗)). Notice that 𝐽(𝜙^∗,0)=inf_{(𝜙,𝜓)∈𝑁}𝐽(𝜙,𝜓), 𝐽(𝜙^∗,0)=(𝑎₁/4)∑_𝑛∈ℤ(𝜙^∗_𝑛)⁴, and 𝐻₂𝜙^∗,0=𝐻₁𝜙^∗,0=𝐿₁−𝜔₁𝜙^∗,𝜙^∗≥𝜃₁₁−𝜔₁𝜙^∗2. (75) For the sake of simplicity, we let 𝐵=_𝑛∈ℤ𝜙^∗_𝑛⁴,𝐷=𝐿₂−𝜔₂𝜙^∗,𝜙^∗. (76) If 𝜔₁≤𝜔₂, then, by (28) and (75), 𝐷=𝐿₂−𝜔₂𝜙^∗,𝜙^∗=𝐿₁−𝜔₁𝜙^∗,𝜙^∗+𝐿₂−𝐿₁𝜙^∗,𝜙^∗+𝜔₁−𝜔₂𝜙^∗,𝜙^∗≤𝑎₁𝐵+𝜆𝜙^∗2≤𝜆+𝜃₁₁−𝜔₁𝜃₁₁−𝜔₁𝑎₁𝐵. (77) This, combined with 𝑎₃>((𝜆+𝜃₁₁−𝜔₁)/(𝜃₁₁−𝜔₁))𝑎₁ and (77) yields 𝑎₁𝐵𝐷<𝑎₁𝑎₃𝐵². If 𝜔₂<𝜔₁, then, by (28) and (75), 𝐷=𝐿₂−𝜔₂𝜙^∗,𝜙^∗=𝐿₁−𝜔₁𝜙^∗,𝜙^∗+𝐿₂−𝐿₁𝜙^∗,𝜙^∗+𝜔₁−𝜔₂𝜙^∗,𝜙^∗=𝑎₁𝐵+𝜆𝜙^∗2+𝜔₁−𝜔₂𝜙^∗2≤𝑎₁𝐵+𝜆+𝜔₁−𝜔₂𝜃₁₁−𝜔₁𝑎₁𝐵=𝜆+𝜃₁₁−𝜔₂𝜃₁₁−𝜔₁𝑎₁𝐵. (78) This, together with 𝑎₃>((𝜆+𝜃₁₁−𝜔₂)/(𝜃₁₁−𝜔₁))𝑎₁ and (78) gives 𝑎₁𝐵𝐷<𝑎₁𝑎₃𝐵². Thus, if 𝑎₃>max{((𝜆+𝜃₁₁−𝜔₁)/(𝜃₁₁−𝜔₁))𝑎₁,((𝜆+𝜃₁₁−𝜔₂)/(𝜃₁₁−𝜔₁))𝑎₁}, then we have 𝑎₁𝐵𝐷<𝑎₁𝑎₃𝐵². For 𝜖 small enough, we have 𝐻²₁𝜙^∗,𝜖𝜙^∗=𝐿₁−𝜔₁𝜙^∗,𝜙^∗+𝐿₂−𝜔₂𝜖𝜙^∗,𝜖𝜙^∗2=𝑎₁𝐵+𝜖²𝐷²=𝑎²₁𝐵²+2𝑎₁𝐵𝐷𝜖²+𝐷²𝜖⁴<𝑎²₁𝐵²+2𝑎₁𝑎₃𝐵²𝜖²+𝑎₁𝑎₂𝐵²𝜖⁴=𝑎₁𝐵𝑎₁𝐵+𝑎₂𝐵𝜖⁴+2𝑎₃𝐵𝜖²=𝐻₂𝜙^∗,0𝐻₂𝜙^∗,𝜖𝜙^∗. (79) Hence, by (79), we have 𝐽𝑡^∗𝜙^∗,𝑡^∗𝜖𝜙^∗=𝐻²₁𝜙^∗,𝜖𝜙^∗4𝐻₂𝜙^∗,𝜖𝜙^∗<14𝐻₂𝜙^∗𝜙,0=𝐽^∗,0=inf_{(𝜙,𝜓)∈𝑁}𝐽𝜙,𝜓. (80) This is a contradiction. So, 𝜓^∗≠0. The proof is completed.

Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable suggestions. 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), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and SRF of Guangzhou Education Bureau (no. 10A012).