Discrete Schrödinger equations in the nonperiodic and superlinear cases: homoclinic solutions

Advances in Difference Equations, Sep 2017

Using variational methods, we study the existence and multiplicity of homoclinic solutions for a class of discrete Schrödinger equations in infinite m-dimensional lattices with nonlinearities being superlinear at infinity. Our results generalize some existing results in the literature by using some weaker conditions.

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Discrete Schrödinger equations in the nonperiodic and superlinear cases: homoclinic solutions

Jia et al. Advances in Difference Equations Discrete Schrödinger equations in the nonperiodic and superlinear cases: homoclinic solutions Liqian Jia 0 1 2 3 5 Jun Chen 0 1 2 4 Guanwei Chen 0 1 2 3 5 0 R is the discrete Laplace operator in the m-dimensional space, ω ∈ , V = 1 Province 250022 , P.R. China 2 University of Jinan , Jinan, Shandong 3 School of Mathematical Sciences 4 Sun Yueqi Honors College, China University Of Mining And Technology , Xuzhou, Jiangsu Province 221116 , P.R. China 5 School of Mathematical Sciences, University of Jinan , Jinan, Shandong Province 250022 , P.R. China Using variational methods, we study the existence and multiplicity of homoclinic solutions for a class of discrete Schrödinger equations in infinite m-dimensional lattices with nonlinearities being superlinear at infinity. Our results generalize some existing results in the literature by using some weaker conditions. discrete nonlinear Schrödinger equations; variational methods - Superlinear; homoclinic solutions 1 Introduction and main results The discrete nonlinear Schrödinger equation is a very important discrete model, which has many important applications in many fields, such as nonlinear optics [], biomolecular chains [], Bose-Einstein condensates [], and so on. In general, discrete nonlinear Schrödinger equation can be divided into two different cases, the periodic and nonperiodic cases. So far, most results are all about the periodic cases, such as [–] and so on. Only a few results are about the nonperiodic cases, such as [–]; in paticular, the papers [, , , , ] are only about the case of onedimensional lattice (n ∈ Z). Inspired by the papers mentioned, we study homoclinic solutions (lim|n|=|n|+|n|+···+|nm|→∞ un = ) of the following nonperiodic discrete nonlinear equation: – un + vnun – ωun = fn(un), n ∈ Zm, (.) where un = u(n+,n,...,nm) + u(n,n+,...,nm) + · · · + u(n,n,...,nm+) – mu(n,n,...,nm) + u(n–,n,...,nm) + u(n,n–,...,nm) + u(n,n,...,nm–) R × . Problem (.) comes from the study of standing waves for the discrete nonlinear Schrödinger equation iψ˙ n = – ψn + vnψn – fn(ψn), n ∈ Zm. Clearly, (.) becomes (.) by the definition of standing waves (ψn = une–iωt with lim|n|→∞ un = ). Therefore, the problem on the existence of standing waves of (.) reduces to that on the existence of homoclinic solutions of (.). We use some suitable assumptions to overcome the difficulties caused by the unboundedness of Zm and the lack of periodic conditions. In particular, condition (F) is a new superquadratic condition introduced by Tang and Wu []. (V) V = {vn}n∈Zm satisfies |n|l→im+∞ vn = +∞. (F) fn ∈ C(R, R), lim|s|→ s fn(s) = . (F) lim|s|→+∞ F|ns|(s) = +∞ for all n ∈ Zm, where Fn(s) := s fn(t) dt, (n, s) ∈ Zm × R. (F) Fn(s) = fn(s)s – Fn(s) ≥  for all (n, s) ∈ Zm × R, and there exist b >  and r∞ >  such that Fn(s) ≥ b F|ns|(s) , ∀n ∈ Zm, ∀|s| ≥ r∞. (F) Fn() ≡  for all n ∈ Zm. (F) There is L >  such that supn∈Zm,|s|=r∞ Fn(s) ≤ L. Theorem . Equation (.) possesses at least one nontrivial homoclinic solution u if conditions (V) and (F)-(F) hold. Here, u is nontrivial, that is, un ≡ . Theorem . Equation (.) has infinitely many nontrivial homoclinic solutions if conditions (V) and (F)-(F) hold and fn(–s) = –fn(s) for all (n, s) ∈ Zm × R. Example . We give the following example to explain the rationality of the assumptions for the nonlinear terms fn. Let Fn(s) = ⎨⎧ an[c|s| + c|s|], |s| ≤ , ⎩ an[|s| ln( + |s|) + sin |s| – ln( + |s|)], |s| > , where s ∈ R,  < infn∈Zm an < supn∈Zm an < +∞, and c, c >  are two suitable constants. It is not hard to check that it satisfies our conditions (F)-(F). Remark . (Comparisons) We give detailed comparisons between our results and the results [–, , , , ] for infinite m-dimensional lattices. Our Theorems . and . generalize the results mentioned. (.) (.) (.) () The results of [, , , , , ] are all based on conditions (V) and (F) fn ∈ C(R, R), lim|s|→ s fn(s) = , and there are a >  and ν >  such that () The authors of [, ] used the Ambrosetti-Rabinowitz superlinear condition fn(s) ≤ a  + |s|ν– , ∀(n, s) ∈ Zm × R. However, we remove condition (.).  < ν Fn(s) ≤ fn(s)s for some ν > , ∀s ∈ R \ {}. Obviously, (.) is stronger than our condition (F). Besides, the authors of [, , ] used the condition fn(s) s is increasing for s >  and decreasing for s < , the authors of [] used the condition lim inf fn(s)s – Fn(s) ≥ b for some b > , > max{, ν – }, |s|→+∞ |s| ∀n ∈ Zm, (.) and the authors of [] used the condition μFn(s) ≤ fn(s)s + κs, μ > , κ > , ∀(n, s) ∈ Zm × R. It is not hard to check that the functions in our Example . do not satisfy conditions (.)-(.), but they all satisfy our conditions (F)-(F). Therefore, our results extend those in the papers mentioned. In Section , we establish the variational framework of (.) and give some preliminary lemmas. In Sections  and , we give detailed proofs of Theorems . and ., respectively. lp ≡ lp Zm := u = {un} : n ∈ Zm, un ∈ R, u lp = n∈Zm |un|p /p < ∞ , p ∈ [, +∞), be real sequence spaces. The following elementary embedding relations hold: lp ⊂ lq, u lq ≤ u lp ,  ≤ p ≤ q ≤ ∞, where u l∞ := nm∈Zaxm |un|. Let L := – + V be defined by Lun := – un + vnun for u ∈ l. Let E be the form domain of L, that is, E := D(L/) (the domain of L/). Under our assumptions, the operator L is an unbounded self-adjoint operator in l. Since the operator – is bounded in l, it is easy to (.) (.) (.) (.) see that E = {u ∈ l : V /u ∈ l}, where V /u is defined by V /un := vn/un for u ∈ l. We define respectively on E the inner product and norm by (u, v)E := (u, v)l + L/u, L/v l and u E = (u, u)E/, where (u, v)l is the inner product in l. Then E is a Hilbert space. Lemma . ([]) If (.) holds, then we have: () The embedding maps from E into lp are compact for all p ∈ [, ∞], and there exist γq >  such that u lq ≤ γq u , ∀u ∈ E. () The spectrum σ (L – ω) consists of the eigenvalues: λ – ω < λ – ω < · · · < λk – ω < · · · → +∞. Let ek be the eigenfunctions with (L – ω)ek = (λk – ω)ek and ek l = , k = , , . . . . Moreover, {ek : k = , , . . .} is an orthonormal basis of l. Let (D) denote the number of i such that i ∈ D. Let and k := k := {i : λi – ω < } , {i : λi – ω = } , k := k + k E– := span{e, . . . , ek }, E := span{ek+, . . . , ek }, E+ := span{ek+, . . .}, E = E– ⊕ E ⊕ E+ where the closure is taken with respect to the norm decomposition · E. Then we have the orthogonal with respect to the inner product (·, ·)E. Now, we introduce respectively on E the following inner product and norm:   (u, v) := u, v l + L  u, L  v l ,  u = (u, u)  , where u, v ∈ E = E– ⊕ E ⊕ E+ with u = u– + u + u+ and v = v– + v + v+. Clearly, the norms · and · E are equivalent, and the decomposition E = E– ⊕ E ⊕ E+ is also orthogonal with respect to both inner products (·, ·) and (·, ·)l . (.) In view of the above arguments, we consider the functional on E defined by (u) =  (L – ω)u, u l – =  u+  –  u–  – n∈Zm Fn(un) n∈Zm Fn(un). Under our assumptions, ∈ C(E, R), and the derivative is given by (u), v = u+, v+ – u–, v– – fn(un)vn,  + uk → , k → ∞, where u, v ∈ E = E– ⊕ E ⊕ E+ with u = u– + u + u+ and v = v– + v + v+. The standard argument shows that nonzero critical points of are nontrivial solutions of (.). Definition . We say that () I ∈ C(X, R) satisfies (C)-condition if any sequence {uk} such that I(uk) is bounded and has a convergent subsequence. () I ∈ C(X, R) satisfies (PS)-condition if any sequence {uk} such that I(uk) is bounded and (.) (.) (.) I uk → , k → ∞, has a convergent subsequence. We shall use the following two lemmas to prove our main results: Lemma . ([]) Let E be a real Banach space, and let I ∈ C(E, R) satisfy (PS)-condition. Suppose I() =  and () there are constants ρ, α >  such that I|∂Bρ ≥ α; () there is e ∈ E \ Bρ such that I(e) ≤ . Then I possesses a critical value c ≥ α. Moreover, c can be characterized as c = infg∈ maxu∈g([,]) I(u), where = {g ∈ C([, ], E) | g() = , g() = e}. Lemma . ([]) Let X be an infinite-dimensional Banach space such that X = Y ⊕ Z, where Y is finite-dimensional. Let I ∈ C(X, R) be an invariant functional. Suppose that, for any k ∈ N , there exist ρk > rk >  such that () I satisfies (C)-condition for all c > ; () ak := maxu∈Yk, u =ρk I(u) ≤ ; () bk := infu∈Yk, u =ρk I(u) → ∞, k → ∞. Then I has an unbounded sequence of critical values. Let {ej}j∞= be an orthonormal basis of E, and let Xj := Rej. Then Yk = span{e, . . . , ek} and Zk = j∞=k Xj = span{ek, . . .} for all k ∈ N. jk= Xj = Lemma . If assumptions (V) and (F)-(F) hold, then satisfies (C)-condition. Proof We assume that, for any sequence {uk} ⊂ E, (uk) is bounded and uk ) → . Then there exists a constant M >  such that uk ≤ M, uk uu(kki).FTirhsetn,wevkpr=ove. Wthee cbaonunasdseudmneestshoatf v{ukk}. Ivf =no{vt,n}thn∈eZnm iunkE →pas∞sinagstok a→su∞bs.eLqeutenvkce=, which, together with Lemma ., implies vk → v in lq for  ≤ q < ∞ and vkn → vn for all n ∈ Zm. By the space decomposition we have u  = u+  + u– + u . Then, by (.), (.), and (.) we have n∈Zm n∈Zm which implies that, for k large enough, we have Fnu(kukn) ≤  + uMk  ≤ . Fnu(kukn) =  – (uk)  (uk)–  + (uk)– + (uk)  uk  –  uk  , If v = , then we let A := {n ∈ Zm : |vn| > }. For all n ∈ A, by vkn = uuknk and uk → ∞ we have limk→∞ |ukn| = ∞. It follows from (F) that Fn(s) ≥  for all (n, s) ∈ Zm × R (see AX  in Appendix) and the Fatou lemma that lim k→∞ n∈Zm Fn(ukn) uk  ≥ lim k→∞ n∈A = kl→im∞ n∈A F|nu(knu|kn) vkn  = +∞, (uk) ( + (.) (.) (.) (.) (.) which contradicts with (.). So, in this case, {uk} is bounded in E. If v = , then vk →  in lq,  ≤ q < ∞, and vkn →  for all n ∈ Zm. Since dim(E– ⊕ E) < ∞, it follows from (.) and (.) that, for k large enough, there exists a constant l >  such that n∈Zm Fnu(kukn) ≥  – uMk  – l n∈Zm =  – uMk  – l n∈Zm  ≥  . |(ukn)–| + |(ukn)– + (ukn)| uk  vkn –  + vkn – + vkn   Then by (F), for any ε > , there exists σ >  such that fn(s) ≤ ε|s|, |s| ≤ σ , n ∈ Zm. It follows from Fn(s) ≥  for all (n, s) ∈ Zm × R (see AX  in Appendix) and (F) that, for all n ∈ Zm and |s| ≤ σ , Let ε = . Then there exists σ >  such that (.) holds for all n ∈ Zm and |s| ≤ σ. By (F) we have ddϑ ϑ –Fn(ϑ s) ≥  for all ϑ ≥ , so ϑ –Fn(ϑ s) is nondecreasing in ϑ for ϑ ≥ . Then by (F), for all |s| ≤ r∞, we have Fn(s) ≤ Fn r|∞s|s |s|  r∞ ≤ Fn r∞s |s| ≤ L. Then since Fn(s) ≥  for all (n, s) ∈ Zm × R (see AX  in Appendix), by (F), (.), (.), (.), (.), and the Sobolev embedding theorem, for k large enough, we have Fn(s) = Fn(s) – Fn() = ≤ ≤       fn(ts)s dt fn(ts) |s| dt ε|ts||s| dt ≤ ε|s|.  ≤ ≤ n∈Zm F|nu(knu|kn) + Fn(ukn)|ukn| σ uk  {n∈Zm,|uk|>r∞} {n∈Zm,|uk|≤σ} Clearly, (.) contradicts with (.). Thus uk is still bounded in this case. (.) (.) (.) (ii) Second, we prove that {uk} has a convergent subsequence in E. The boundedness of {uk} implies that uk u in E+ passing to a subsequence, where u = {un}n∈Zm . Now we have n∈Zm fn ukn ukn – un → , k → ∞. Note that Lemma . implies that uk → u in lq for all  ≤ q < ∞, so we have The boundedness of {uk} and Lemma . imply that uk q < ∞ for all  ≤ q < ∞. Then by (F), (.), and the Hölder inequality, for ε > , there exists δ >  such that, for |s| < δ, we have (.) (.) (.) (.) (.) uk – u l → . n∈Zm fn ukn ukn – un So (.) holds. Therefore, since we have  = lim k→∞ uk , uk – u = lim uk, uk – u – lim k→∞ k→∞ n∈Zm = lim uk  – u  – , k→∞ that is, lim uk = u . k→∞ Since uk u in E+, it follows that ≤ ≤ = ε n∈Zm n∈Zm n∈Zm fn ukn ukn – un ε ukn ukn – un ukn ukn – un ≤ ε uk l uk – u l → . fn ukn ukn – un (uk) → , uk u in E+, by (.) and the definition of uk – u  = uk – u, uk – u → , that is, {uk} has a convergent subsequence in E+. Since dim(E– ⊕ E) < ∞, it follows that {uk} has a convergent subsequence in E. Thus satisfies (C)-condition. 3 Proof of Theorem 1.1 Lemma . If assumptions (V), (F), (F), and (F) hold, then there exist constants , α >  such that |S ≥ α, where S = {u ∈ E+| u = }. Proof In view of (.), let ε = γ , where γ is defined in Lemma .. Then there exists σ >  such that Fn(s) ≤ ε|s| for all n ∈ Zm and |s| ≤ σ. Let ρ = γσ∞ , α =   > ,  where γ∞ is defined in Lemma .. This implies that  < u l∞ ≤ σ for all u ∈ S. Then by (.) we have ≥  u  –  u  =  u . Thus, by (.) and the definitions of and α, the proof of the lemma is finished. Lemma . If assumptions (V) and (F)-(F) hold, then there exists ζ > such that |∂Q ≤  with defined in Lemma .. Proof Let e ∈ E+ with e =  and K = E– ⊕ E ⊕ span{e}. Then there exists a small enough constant ε >  such that n ∈ Zm : |un| ≥ ε u ≥ , ∀u ∈ K \ {}. The detailed proof of (.) is given in the Appendix. For u = u+ + u + u– ∈ K , let u = {n ∈ Zm | |un| ≥ ε u }. By (F), for M = /ε > , there exists L >  such that Fn(s) ≥ M|s|, ∀|s| ≥ L, ∀n ∈ Zm. Thus we have Fn(un) ≥ M|un| ≥ Mε u , ∀n ∈ u, (u) =  u+  –  u–  – n∈Zm ≤  u+  – n∈ u Fn(un) ≤  u+  – Mε u  · ( u) ≤  u+  – Mε u  ≤  where u ∈ K and u ≥ L/ε. It follows from (F) and (.) that (.) (.) (.) for all u ∈ K and u ≥ L/ε. Let Q = {ηe| ≤ η ≤ ζ } ⊕ {u ∈ E– ⊕ E| u ≤ ζ }. Then we have ∂Q = Q ∪ Q ∪ Q, where Q = u ∈ E– ⊕ E | u ≤ ζ , Q = ζ e ⊕ u ∈ E– ⊕ E | u ≤ ζ , Q = {ηe |  ≤ η ≤ ζ } ⊕ u ∈ E– ⊕ E | u = ζ . Then by (.), for all ζ ≥ L/ε, we have (u) ≤  for all u ∈ Q ∪ Q. It follows from (F) that (u) ≤  for all u ∈ E– ⊕ E, which implies that (u) ≤  for all u ∈ Q. Thus, for all ζ > max{ , L/ε}, we have (u) ≤ , ∀u ∈ ∂Q. The proof is finished. Proof of Theorem . Similarly to Lemma ., we can also prove that satisfies (PS)condition. Then by Lemmas . and . conditions () and () of Lemma . hold, so Lemma . implies that possesses a critical point u such that (u) ≥ α. Therefore u is a homoclinic solution of problem (.). 4 Proof of Theorem 1.2 Let βk(p) = sup u∈Zk, u = u lp , k ∈ N , q ∈ [, +∞]. bk = Lemma . If assumptions (V), (F), and (F)-(F) hold, then bk → ∞ as k → ∞, where Proof By Lemma ., E is embedded compactly into lp. Then βk(p) →  as k → ∞. For k large enough, we choose k such that Zk ⊂ E+. Note that τk → +∞ as k → ∞, and so for any u ∈ Zk with u = τk , we have u l∞ ≤ r∞. Fn(un) ≤ |un|. Then by (.), there exists r∞ ≥ σ > , for all |un| ≤ σ, we have (.) (.) Then by (.), (.), (.), and (.), for any u ∈ Zk with u = τk , we have =  u  –  + σL ≥  u  –  + σL u  βk()  u l ≥  u  =  τk, k > k, k ∈ N . n∈Zm,σ≤|un|≤r∞ Fn(un) |uσn| Fn(un)  Then we have bk = (u) ≥  τk → +∞, k → ∞. So the lemma is proved. Lemma . If assumptions (V) and (F) hold, then we have ak ≤  for all k ∈ N , where ak = max u∈Yk, u =ρk (u). δ(u) = n ∈ Zm : |un| ≥ δ u . ε (u) ≥ .  Fn(un) ≥ ε |un| ≥ u .  Proof For any u ∈ Yk/{}, δ > , where u = u– + u + u+, let By (.), for all u ∈ Yk/{}, we obtain that there exists ε >  such that By (F), there exists γ >  such that , for all u ∈ Yk and n ∈ ε (u) with u ≥ γ, we have (.) (.) We choose ρk > max{γ, τk}. It follows from (.), (.), (.), and Fn(s) ≥  for all (n, s) ∈ Zm × R (see AX  in Appendix) that, for any u ∈ Yk with u = ρk , (u) =  u+  –  u–  + n∈Zm ≤  u  – ≤  u  – u  ≤ –  u  ≤ , n∈ ε (u) Fn(un) ε (u) which means that ak ≤  for all k ∈ N , and the proof of the lemma is finished. Proof of Theorem . Let X = E, Y = Ym, and Z = Zm. By Lemma ., under our assumptions, satisfies (C)-condition. Clearly, condition () of Lemma . holds. Besides, conditions () and () of Lemma . hold by Lemmas . and ., respectively. So problem (.) possesses infinitely many nontrivial solutions by Lemma .. Therefore, Theorem . is true. Appendix In this section, we prove the following facts. AX  If assumptions (F), (F), and (F) hold, then Fn(s) ≥  for all (n, s) ∈ Zm × R. Proof g(t) = Fnt(ts) , (n, s) ∈ Zm × R, t > . By condition (F) we have g (t) = fn(ts)ts t– Fn(ts) ≥ , (n, s) ∈ Zm × R, which implies that g(t) is nondecreasing in (, +∞). By (F) and (F) we have lim g(t) = lim Fn(ts) = lim fn(ts)s ≤ lim |fn(ts)| |s| =  t→ t→ t t→ t t→ ts  for all (n, s) ∈ Zm × R/{}. Thus limt→ g(t) = , which shows that g(t) ≥  for all t >  and (n, s) ∈ Zm × R/{}. It follows from (F) that Fn(s) = g() ≥ , (n, s) ∈ Zm × R. Thus the proof is finished. AX  The fact (.) in Lemma . holds. Proof If not, for any positive integer k, there exists uk ∈ K \ {} such that n ∈ Zm : ukn ≥ k uk = . n ∈ Zm : vkn ≥ k = . Let vk = uk/ uk , then vk = , and for all k, we have Since dim K < ∞, we may assume that vk v = {vn}n∈Zm in W passing to a subsequence, and thus v = . By Lemma . we have vk → v in l and vkn → vn for all n ∈ Zm, so that (A.) (A.) · l∞ (A.) vkn – vn  → , k → ∞. n∈Zm Let n ∈ Zm : |vn| ≥ δ ≥ .  = n ∈ Zm | |vn| ≥ δ , k = n ∈ Zm vkn < k , ck = Zm \ k = n ∈ Zm The fact that v =  implies v l∞ = maxn∈Zm |vn| > . By the definition of norm there exists a constant δ >  such that vkn ≥ k . By (A.) and (A.) we have ( k ∩ ) = c  \ k ∩  ≥ ( ) – ck ≥  –  = . Next, we may assume that δ – k ≥  δ for k large enough. Then we have vkn – vn  ≥ vkn – |vn|  ≥ δ – k     δ , ≥   ∀n ∈ k ∩ , and thus, for all large k, we have vkn – vn  ≥ n∈Zm n∈ k∩  vkn – vn  ≥  δ · ( k ∩ ) ≥  δ > . Clearly, it is a contradiction to (A.). Thus (.) holds. Funding Research supported by National Natural Science Foundation of China (No. 11401011, No. 11771182) and Natural Science Foundation of Shandong Province (No. ZR2017JL005). Availability of data and materials Not applicable. Ethics approval and consent to participate Not applicable. Competing interests The authors declare that they have no competing interests. Consent for publication Not applicable. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1. Christodoulides , DN , Lederer, F , Silberberg, Y: Discretizing light behaviour in linear and nonlinear waveguide lattices . Nature 424 , 817 - 823 ( 2003 ) 2. Kopidakis , G , Aubry, S , Tsironis, GP : Targeted energy transfer through discrete breathers in nonlinear systems . Phys. Rev. Lett . 87 , Article ID 165501 ( 2001 ). doi: 10 .1103/PhysRevLett.87.165501 3. Livi , R , Franzosi, R , Oppo, GL : Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation . Phys. Rev. Lett . 97 , Article ID 060401 ( 2006 ). doi: 10 .1103/PhysRevLett.97.060401 4. Chen , G , Ma, S: Discrete nonlinear Schrödinger equations with superlinear nonlinearities . Appl. Math. Comput . 218 , 5496 - 5507 ( 2012 ) 5. Chen , G , Ma, S: Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities . Stud. Appl . Math. 131 , 389 - 413 ( 2013 ) 6. Chen , G , Ma, S: Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms . Appl. Math. Comput . 232 , 787 - 798 ( 2014 ) 7. Chen , G , Ma, S , Wang , Z-Q : Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities . J. Differ. Equ . 261 , 3493 - 3518 ( 2016 ) 8. Pankov , A : Gap solitons in periodic discrete nonlinear equations . Nonlinearity 19 , 27 - 40 ( 2006 ) 9. Pankov , A : Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach . Discrete Contin. Dyn. Syst . 19 , 419 - 430 ( 2007 ) 10. Pankov , A , Rothos , V : Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity . Proc. R. Soc. A 464 , 3219 - 3236 ( 2008 ) 11. Pankov , A : Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities . J. Math. Anal. Appl . 371 , 254 - 265 ( 2010 ) 12. Shi , H , Zhang, H: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations . J. Math. Anal. Appl . 361 , 411 - 419 ( 2010 ) 13. Shi , H: Gap solitons in periodic discrete Schrödinger equations with nonlinearity . Acta Appl . Math. 109 , 1065 - 1075 ( 2010 ) 14. Yang , M , Chen, W , Ding, Y: Solutions for discrete periodic Schrödinger equations with spectrum 0 . Acta Appl . Math. 110 , 1475 - 1488 ( 2010 ) 15. Zhou , Z , Yu, J , Chen, Y: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity . Nonlinearity 23 , 1727 - 1740 ( 2010 ) 16. Zhou , Z , Yu, J , Chen, Y: Homoclinic solutions in periodic difference equations with saturable nonlinearity . Sci. China Math. 54 , 83 - 93 ( 2011 ) 17. Chen , G , Schechter, M: Non-periodic discrete Schrödinger equations: ground state solutions . Z. Angew. Math. Phys. 67 , 1 - 15 ( 2016 ) 18. Jia , L , Chen, G: Multiple solutions of discrete Schrödinger equations with growing potentials . Adv. Differ. Equ . 2016 , Article ID 275 ( 2016 ). doi:10.1186/s13662-016-1003-3 19. Jia , L , Chen, G: Discrete Schrödinger equations with sign-changing nonlinearities: infinitely many homoclinic solutions . J. Math. Anal. Appl . 452 , 568 - 577 ( 2017 ) 20. Ma , D , Zhou , Z: Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials . Abstr. Appl. Anal . 2012 , Article ID 703596 ( 2012 ). doi: 10 .1155/ 2012 /703596 21. Pankov , A , Zhang, G: Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity . J. Math. Sci. 177 , 71 - 82 ( 2011 ) 22. Pankov , A : Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities . Appl. Anal . 92 , 308 - 317 ( 2013 ) 23. Sun , G: On standing wave solutions for discrete nonlinear Schrödinger equations . Abstr. Appl. Anal . 2013 , Article ID 436919 ( 2013 ) 24. Zhang , G , Pankov, A : Standing waves of the discrete nonlinear Schrödinger equations with growing potentials . Commun. Math. Anal. 5 , 38 - 49 ( 2008 ) 25. Zhang , G, Liu, F: Existence of breather solutions of the DNLS equations with unbounded potentials . Nonlinear Anal . 71 , 786 - 792 ( 2009 ) 26. Zhang , G: Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials . J. Math. Phys. 50 , Article ID 013505 ( 2009 ) 27. Zhang , G , Pankov, A : Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, II . Appl. Anal. 89 , 1541 - 1557 ( 2010 ) 28. Zhou , Z , Ma, D: Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials . Sci. China Math. 58 , 781 - 790 ( 2015 ) 29. Tang , CL , Wu, XP : Periodic solutions for a class of new superquadratic second order Hamiltonian systems . J. Math. Anal. Appl . 34 , 65 - 71 ( 2014 ) 30. Rabinowitz , PH : Minimax Methods in Critical Point Theory with Applications to Differential Equations . CBMS Regional Conference Series in Mathematics, vol. 65 . Am. Math. Soc. , Providence ( 1986 ) 31. Willem , M: Minimax theorems . Prog. Nonlinear Differ. Equ. Appl . 50 , 139 - 141 ( 1996 )


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Liqian Jia, Jun Chen, Guanwei Chen. Discrete Schrödinger equations in the nonperiodic and superlinear cases: homoclinic solutions, Advances in Difference Equations, 2017, 289,