Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrödinger equations −(a+b∑k∈Z|Δuk−1|2)Δ2uk−1+Vkuk=fk(uk)k∈Z,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$\end{document} where a, b are two positive constants and V={Vk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V=\{V_{k}\}$\end{document} is a positive potential. Δuk−1=uk−uk−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta u_{k-1}=u_{k}-u_{k-1}$\end{document} and Δ2=Δ(Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta ^{2}=\Delta (\Delta )$\end{document} is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities {fk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{f_{k}\}$\end{document} satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.


Introduction
In the present paper, we consider the following discrete Schrödinger equations where a, b are two positive constants and V = {V k } is a positive potential. u k-1 = u ku k-1 and 2 = ( ) is the one-dimensional discrete Laplacian operator. The discrete Schrödinger equations play a significant role in many areas, such as nonlinear optics [7], biomolecular chains [11] and Bose-Einstein condensates [16]. If a = 1 and b = 0, problem (1) reduces to the classical discrete Schrödinger equations, which have been extensively studied by many authors in the past several decades. In Ma and Guo [17] and Zhang and Pankov [32], the authors studied the nontrivial solution of discrete Schrödinger equations with a coercive potential by variation methods. In Lin et al. [15], the authors considered a class of discrete nonlinear nonperiodic systems and obtained the existence of the homoclinic solutions. For more related works, we refer to [4-6, 18-20, 25, 28] and their references.
The problem (1) is a discrete case for a class of nonlocal problems and we call this nonlocal term a Kirchhoff type. This class of nonlocal equations is an extension of the classi-cal d' Alembert's wave equations because of the effects of the changes in the length of the string during the vibrations. There is a large number of papers concerning the solutions of Kirchhof-type problems in the continuous case, such as [9,12,26,29,31] and their references. As for the discrete equations with a nonlocal term, Yang and Liu in [30] investigated the existence of the nontrivial solutions via critical groups. In [3], Chakrone et al. considered the multiplicity results for a p-Laplacian discrete problem with Neumann boundary conditions. For more related works, we refer to [1,2,8,13,14,[22][23][24]34] and their references. Recently, Wu and Huang [27] have studied the statistical solutions for nonautonomous discrete Klein-Gordon-Schrödinger-type equations.
Most of the previous works have been concerned with boundary value problems, while little has been done in discrete Schrödinger equations of the Kirchhoff type. We will consider the existence of infinitely many nontrivial solutions for this class of discrete equations. In this paper, the potential V satisfies the coercive condition: then L is a self-adjoint operator and unbounded in l 2 . With the help of the condition (V 1 ), the authors in [32] proved that the spectrum σ (L) is discrete going to +∞ and we assume that λ i are the eigenvalues of L. If we set that then E is a Hilbert space, in which the norm and the inner product are defined by respectively. Denote the standard norms of l q by · q for q ∈ [2, +∞]. First, we consider the 4-superlinear nonlinearities as follows: (f 1 ) f k ∈ C(R, R) and there exist C > 0 and p > 2 such that f k (t) ≤ C |t| + |t| p-1 for any k ∈ Z and t ∈ R.
f k (t)t + αt 2 for any k ∈ Z and t ∈ R.
(f 3 ) There exists L > 0 such that f k (t)t for any k ∈ Z and |t| ≥ L.
(f 4 ) f k (-t) = -f k (t) for any k ∈ Z and t ∈ R. If u = {u k } satisfies problem (1) for any k ∈ Z, we call that u a solution of this problem. Moreover, if u = 0, we call that u a nontrivial solution. With the above assumptions (V 1 ) and (f 1 ), the energy functional I: E → R is well defined by Moreover, I is a class of C 1 (E, R) and It is evident to check that (f 1 ) and (f 3 ) imply (f 3 ). Hence, we have the following result.
The above two results obtained infinitely many high-energy solutions are strictly dependent on the 4-superlinear growth assumption. The standard Symmetric Mountain Pass Theorem may be invalid without the superlinear assumption. In this case, we try to establish the existence of infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya in [10] when the nonlinearities f k (t) are sublinear at infinity. More precisely, we assume that In fact, we will prove a more general result than the above theorem.

The 4-superlinear case
The following Symmetric Mountain Pass Theorem is crucial in proving the existence of infinitely many solutions with 4-superlinear nonlinearities.

Proposition 2.1 (Symmetric Mountain Pass Theorem [21]) Let X be an infinite
satisfies the (Ce) condition, and

then J possesses an unbounded sequence of critical values.
In [21], the Symmetric Mountain Pass Theorem is established under the (PS) condition.
Since the Deformation Theorem is still valid under the (Ce) condition, we see that the Symmetric Mountain Pass Theorem also holds under the (Ce) condition. The following embedding lemma, which follows from [17] or [33], plays a significant role in recovering the compactness result. which implies that 1 4 min{a, 1} ≤ 0. That is impossible. If v = 0, then 1 := {k ∈ Z|v k = 0} = ∅. For any k ∈ 1 , we have |u n k | → +∞ as n → ∞. By (f 2 ), one obtains that for any k ∈ 1 , combined with Fatou's Lemma, which implies that It follows from (f 2 ) that there exists L 1 > 0 such that F k (t) ≥ 0 for any k ∈ Z and |t| ≥ L 1 .
By (f 1 ), we obtain |F k (t)| ≤ Ct 2 for any k ∈ Z and |t| ≤ L 1 . Combining with (3), we have F k (t) ≥ -Ct 2 for any k ∈ Z and t ∈ R.
Hence, we obtain Note that Dividing by u n 4 on both sides and letting n → ∞, we obtain (2) and (4), which is impossible. In any case, we obtain a contradiction and hence {u n } is bounded in E.
Moving if necessary to a subsequence, we can assume u n u in E. It follows that One has min{a, 1} u nu 2 ≤ I u n -I (u), u nu By the boundedness of {u n } and u n u in E, it is obvious that I n u n -I (u), u nu → 0 as n → ∞.
By (f 1 ), Lemma 2.1 and Lebesgue's dominated convergence theorem k∈Z f k u n kf k (u k ) u n ku k → 0 as n → ∞.
Let us consider the functional P : E → R, Since |P(w)| ≤ u w , we can deduce that P is a continuous linear functional on E. By u n u in E, we obtain P u nu = k∈Z u k-1 u n k-1u k-1 → 0 as n → ∞.

By the boundedness of {u
It follows from (5)-(8) that u nu → 0 as n → ∞. Thus, u n → u strongly in E as n → ∞.
Let {e j } be an orthonormal basis of E and define X j = span{e j }, Y m = m j=1 X j and Z m = ∞ j=m+1 X j for any m ∈ N.

Lemma 2.3
Under the assumption (V 1 ), for any 2 ≤ q ≤ +∞, Proof It is obvious that 0 < β m+1 (q) ≤ β m (q), so that β m (q) → β(q) ≥ 0 as m → ∞. For every m ∈ N, there exists u m ∈ Z m with u m = 1 such that For any w ∈ E, w = ∞ j=1 c j e j , by the Cauchy-Schwarz inequality, we have as m → ∞, which implies that u m 0 in E. The compact embedding of E → l q , q ∈ [2, ∞], implies that u m → 0 in l q . Let m → ∞ in (10) and we obtain β m (q) → 0 as m → ∞.
The proof of Theorem 1.1 We will make use of the Symmetric Mountain Pass Theorem and Proposition 2.1 to prove Theorem 1.1. It is easy to see that (J 1 ) follows from the condition (f 4 ). It follows from (f 1 ) that F k (t) ≤ c 1 t 2 + c 2 |t| p for any k ∈ Z and t ∈ R. It follows from the above three inequalities that It follows from p > 2 that there exist δ 1 , α > 0 such that I| ∂B δ 1 ∩Z m ≥ α. Thus, (J 2 ) holds. It remains to prove (J 3 ). Since all norms are equivalent in a finite-dimensional space, there exists c 4 such that u 4 4 ≥ c 4 u 4 for any u ∈ E. By (f 2 ), for any M > b 4c 4 , there exists L 1 > 0 such that F k (t) ≥ Mt 4 for |t| ≥ L 1 . It follows from (f 1 ) that there exists C 1 > 0 such that F k (t) ≥ -C 1 t 2 for |t| ≤ L 1 . From the above two inequalities, it follows that for any k ∈ Z, where C M = C 1 + ML 2 1 . It follows that for all u ∈ E. Hence, there exists a large R = R( E) such that I(u) ≤ 0 on E \ B R . This completes the proof.

The proof of Theorem 1.4
To deal with the sublinear case in Theorem 1.4, we need the following Symmetric Mountain Pass Theorem of Kajikiya [10]. Let X be a Banach space and A be a subset of X. Denote by the family of closed symmetric subsets A of X, = A ∈ X\{0}|A is closed and symmetric with the origin .
where κ and ξ k are given in (f 6 ).
Proof Arguing indirectly, if not, for any positive integer n there exist u n ∈ E such that meas k ∈ Z|ξ k u n k κ ≥ 1 n u n κ = 0.
By the above inequality, we can assume that u n = 1 and By the compactness of the unit sphere of the finite-dimensional subspace E, there exists a subsequence such that u n → u 0 in E and u n → u 0 in l 2 . By the Hölder inequality, we have k∈Z ξ k u n ku 0 κ ≤ k∈Z |ξ k | Since u 0 = 1, there exists d 1 > 0 such that meas{k ∈ Z|ξ k |u 0 k | κ ≥ d 1 } ≥ 1. Otherwise, one has meas{k ∈ Z|ξ k |u 0 k | κ ≥ 1 n } = 0, which implies that 0 ≤ k∈Z ξ k u 0 κ+2 ≤ 1 n k∈Z u 0 k 2 → 0 as n → ∞.