Standing Waves Solutions for the Discrete Schrödinger Equations with Resonance

In this paper, by using linking methods, we obtain the existence of the nontrivial standing wave solutions for the discrete nonlinear Schrödinger equations with resonance and unbounded potentials. In order to prove the existence of standing wave solutions, we give resonant condition to find a bounded critical sequence, and we show that such a sequence guarantees the existence of one nontrivial standing wave solution in l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{2}$$\end{document} when the nonlinearity is resonant and the potential is unbounded. To the best of the our knowledge, there is no existence results for the discrete nonlinear Schrödinger equations with resonance in the literature.


Introduction
In recent years, we can see the nonlinear difference equations have been widely studied and applied by many research fields [1][2][3][4][5][6][7][8][9]. In particular, the discrete nonlinear Schrödinger equations have been studied very intensively in some related problems  of physical and biological phenomena. For example, they can be used to describe how hight intensity light interacts with and propagates through matter [10], biomolecular chains [11], the self-focusing of beams and Bose-Einstein condensates [12], etc. Great effort has been made to study the excellent results for the discrete Schrödinger equations in the literature, such as the existence of gap solitons [13][14][15][16], the chaotic dynamics [17], and ground state solutions [18], etc.
In this paper, we shall study the standing wave problems for the following discrete nonlinear Schrödinger equation: where ψ n = ψ n+1 + ψ n−1 − 2ψ n is the discrete Laplacian in one spatial dimension, the discrete potential V = {v n } is a sequence of real numbers. The nonlinearity f n (u) is continuous in u and f n (0) = 0 for every n ∈ Z. Since f n (0) = 0, then u = 0 is a trivial solution of problem (1).
In problem (1), the nonlinearity f n (·) is supposed to be gauge invariant, i.e., f n e iθ u = e iθ f n (u), θ ∈ R.
We consider the special solutions of problem (1) of the form where ω ∈ R is the temporal frequency and {u n } is a real valued sequence satisfying lim n→±∞ u n = 0.
Such solutions are called standing wave solutions, inserting the standing wave Ansatz into problem (1), and thus, we get the following nonlinear difference equation: with the boundary condition (4). As usual, we also say that a solution of the equation (5) is homoclinic (to 0), that is, the standing wave solutions of problem (1) correspond to the homoclinic solutions of the equation (5) [19][20][21][22]. In particular, when the temporal frequency ω belong to a spectral gap, a solution of the equation (5) is called a gap soliton, we refer readers to [13] for additional details.
In the past decade, most of the literature is devoted to the existence of the standing waves solutions for the discrete nonlinear Schrödinger equations with the superlinear or saturable nonlinearities. Pankov and Chen in [13,22] considered spatially localized standing waves for the problem (1) with the superlinear nonlinearity function f n (u) = |u| 2 u. the superlinear nonlinearity has been considered very intensively in the physics fields [14]. In 2010, Zhou [23,24] considered the standing waves solutions for the problem (1) with typical representatives of saturable nonlinearities f n (u) = 1 − e −c n u 2 u and f n (u) = u 3 1 + c n u 2 −1 , respectively. The saturable nonlinearities can describe optical pulse propagation in various doped fibres [25,26]. The interesting results of other standing wave solutions can be found in various models of nonlinear systems [27][28][29].
In addition, since some difficulties appeared in the process of proof, according to the lack of compactness on the set Z. Most of authors considered to assume the potential V = {v n } is real valued periodic sequence based on the above difficulties [13,24,30,31], and such periodic case overcame these difficulties by means of the critical point theory in combination with periodic approximations. In recent years, there seems to be only a few papers discussed the existence of standing wave solutions for the discrete nonlinear Schrödinger equations with non-periodic coefficients and unbounded potentials, they can be found in the literature [27][28][29]32]. The interesting results were dependent on a discrete version of compact embedding theorem [27]. Resonance is not only used in physics with a very high frequency, but also resonance phenomenon is also one of the most common and frequent natural phenomena in our life [33][34][35]. Everything we see with our eyes are receive signals according to the principle of resonance. Such as we watch TV and surf the internet, etc. To the best of our knowledge, most of these papers dealt with the superlinear or saturable nonlinearities without resonance. Motivated by the above work, we study the nonlinearity which satisfies the resonant case lim |t|→∞ f n (t)/t = λ k − ω and the potential V = {v n } is non-periodic and unbounded in this paper. We shall give sufficient conditions to prove the existence of standing wave solutions for the equation (1) with resonance, where it is vital to find a bounded critical sequence by means of linking methods rather than periodic approximations.

Preliminaries
In order to establish the variational framework associated with the problem (1) and apply the critical point theory, we give some basic notations and lemmas which will be used to prove our main results. We consider the real sequence spaces The following elementary embedding relation holds We give the following assumption (i) The discrete potential V = {v n } satisfies lim n→±∞ v n = +∞.
Without loss of generality, we assume v n ≥ 1 for every n ∈ Z, and we also need the stronger assumption This is a Hilbert space. We also define a norm Since the operator − is bounded, the following two norms are equivalent in E, n u n }. Firstly, we give the following two lemmas which will play an important role in the proof of our main results.

Lemma 1 [27] If V satisfies the condition (i), then
From Lemma 1, we can assume that On the Hilbert space E, we define the C 1 -functional J as follows: where (·, ·) is the inner product in l 2 , and F n (t) = t 0 f n (s)ds for all n ∈ Z. We can compute the Frećhet derivative as follows: It is clear that the critical points of J in E are exactly solutions of the equation (5). Let X be a real Banach space. B ρ is the open ball of radius ρ and center 0 in X and let ∂ B ρ denote its boundary. We give the Chapter II, Theorem 2.7.3 [36] as follows.
Lemma 2 [36] Let M, N are closed subspaces of X such that If m R is bounded, this implies a bound on c.

Main results
In this section, we will look for one standing wave solution of the equation (5) in E. Now, we give some basic assumptions on the nonlinearity f n (·): (ii) f n (t) = o(t) as t → 0, uniformly for all n ∈ Z; (iii) there exists p ∈ (2, +∞) such that f n (t) = o |t| p−1 as |t| → ∞, uniformly for all n ∈ Z.

Proof of Lemma 3
Since u ( j) E is bounded, there is a renamed subsequence {u ( j) } which converges weakly to some u in E. From Lemma 1, we have For such a subsequence, we verify that By (iii), let ε > 0, we take r sufficiently large such that and where w r (t) is a continuous function defined by By combining (11) with the boundedness of the sequence u ( j) E , the first term on the right hand side can be estimated by Similarly we may consider the third term. Moreover, for every n ∈ Z, we notice that f n (t) and w r (t) are continuous, we have From the assumptions (ii) − (iii), it follows easily that for any ε > 0 there exists Then By dominated convergence we obtain Hence (9) holds. Next, we show (10). We write In view of (9), we only need to show By the Minkowski inequality, we have By Lemma 1, we obtain Similarly to show that which show (10) holds.
For any v ∈ E, we see, from (7) and (9), that Hence u is a solution of the second equation in (8).
n → 0, (14) we have, by (10), This implies that u ( j) → u strongly in E, thus J (u) = c. The proof is complete.
When lim |t|→∞ f n (t)/t = λ k − ω, we say the problem (1) is resonant at infinity, where λ k is an eigenvalue of the operator H . Let g n (t) = f n (t) − (λ k − ω)t. We denote G n (t) = t 0 g n (s)ds and consider the following assumptions: (iv) lim |t|→∞ f n (t)/t = λ k − ω > 0 uniformly for all n ∈ Z; (v) There exists some δ > 0 such that {sup |t|<δ |G n (t)|} +∞ n=−∞ ∈ l 1 and G n (t) ≥ 0 for all n ∈ Z and |t| ≥ δ. Moreover, for all n ∈ Z and t ∈ R, we assume that tg n (t) − 2G n (t) ≤ 0 and Example 1 Before proceeding further, we first give a function satisfying the conditions (ii) − (vi) below. Put λ k − ω = 2 ln 2 − √ 2 2 > 0 and δ is taken sufficiently small. The function is defined by Then, Let E(λ k ) be the finite dimensional eigenspace corresponding to the eigenvalue λ k , and let M denote the subspace of E spanned by the eigenfunctions corresponding to the eigenvalues which are greater than λ k .

Lemma 4
Assume that the conditons (i) − (iv) hold, the following alternative hold: M.

Proof of Lemma 4
Under the conditions (ii) and (iv), we get Thus, there is a positive constant δ > 0 such that Let u = z + w, z ∈ E(λ k ), and w ∈ M. Then For z ∈ E(λ k ), we can find a ρ > 0 so small that We assume that u ∈ E(λ k ) M satisfies u E ≤ ρ and |u n | ≥ δ (18) for some n ∈ Z. Then for those n ∈ Z satisfying (18) we have δ ≤ |u n | ≤ |z n | + |w n | ≤ δ 2 + |w n |.
Hence, |z n | ≤ δ 2 ≤ |w n |, |u n | ≤ 2|w n |. In view of (16), (17) and (13), we let ε = λ k+1 −λ k It follows from the elementary embedding relation that Thus, Now we assume that (b) is not true. Then there is a sequence If ρ is sufficiently small, then (19) implies w ( j) E → 0. Consequently, z ( j) E → ρ. We note that E(λ k ) is finite dimensional. We can take a renamed subsequence {z ( j) } such that z ( j) →z ∈ E(λ k ), and we obtain Further, from (16), we have By the definition J and (20), Thus, by (21), we get Let ν be any element in E. Then for t > 0 sufficiently small, we have Taking the limit as t → 0, we see that This completes the proof. Now, we state our main results.

Theorem 1 Assume that (i) − (vi) and (i ) hold, then problem (5) has at least one nontrivial solution u 0 ∈ E.
Proof of Theorem 1 We will use Lemma 2, Lemma 3 and Lemma 4 to prove our results. Let N denote the finite dimensional subspace of E spanned by the eigenfunctions corresponding to the eigenvalues which is less than λ k . Thus we have By (vi), we have On the other hand, for ∀u ∈ E(λ k ) M, if the result (a) of Lemma 4 holds, then problem (5) has at least one nontrivial u 0 =z ∈ E(λ k ). If not, we have, for ρ sufficiently small, for some positive ε 1 . Let e k be an eigenfunction corresponding to λ k such that From (v), there exists φ ∈ l 1 such that |G n (t)| ≤ φ n for all n ∈ Z and |t| < δ, then Let m R = φ l 1 . Obviously, m R /R → 0(R → ∞). It follows from Lemma 2 and Remark 1 that there exists a sequence {u ( j) } ⊂ E satisfying Next, we will show that {u ( j) } is bounded. If not, that is ρ j . Then ũ ( j) E = 1. Consequently, there is renamed subsequence such thatũ ( j) →ũ weakly in E, strongly in l 2 andũ ( j) n →ũ n as j → ∞ for all n ∈ Z.
In view of (24), (25) and (v), we see that This is a contradiction. Thus the critical sequence {u ( j) } is bounded. By (22) and Lemma 3,problem (5) possesses at least one nontrivial solution u in E. This completes the proof.
Author Contributions ZW proposed the idea of this paper and performed all the steps of the proofs. QL wrote the whole paper. All authors read and approved the final manuscript. Availability of data and materials Not applicable.

Conflict of interest
The authors declare that they have no competing interests.
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