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 mdimensional 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 mdimensional 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 [], BoseEinstein 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
(limn=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
limn→∞ 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
nl→im+∞ vn = +∞.
(F) fn ∈ C(R, R), lims→ s
fn(s) = .
(F) lims→+∞ Fns(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 Fns(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[cs + cs], 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 mdimensional lattices. Our Theorems . and
. generalize the results mentioned.
(.)
(.)
(.)
() The results of [, , , , , ] are all based on conditions (V) and
(F) fn ∈ C(R, R), lims→ s
fn(s) = , and there are a > and ν > such that
() The authors of [, ] used the AmbrosettiRabinowitz superlinear condition
fn(s) ≤ a + sν– ,
∀(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
unp
/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 selfadjoint 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 := vn/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 infinitedimensional Banach space such that X = Y ⊕ Z,
where Y is finitedimensional. 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=ove. 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 Fnu(knukn) 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) ≤ εs, 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∞ss
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
εtss dt ≤ εs.
≤
≤
n∈Zm
Fnu(knukn) +
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) ≤ εs 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) ≥ Ms,
∀s ≥ L, ∀n ∈ Zm.
Thus we have
Fn(un) ≥ Mun ≥ 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.
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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 : Selflocalization of BoseEinstein 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 , ZQ : 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: Nonperiodic 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/s1366201610033
19. Jia , L , Chen, G: Discrete Schrödinger equations with signchanging 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: signchanging 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 nonlinear 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 )