Multiple solutions of discrete Schrödinger equations with growing potentials

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R E S E A R C H

Open Access

Multiple solutions of discrete Schrödinger

equations with growing potentials

Liqian Jia and Guanwei Chen

*

*_{Correspondence:} [email protected]

School of Mathematical Sciences, University of Jinan, Jinan, Shandong Province 250022, P.R. China

Abstract

Under some weaker conditions than elsewhere, we obtain infinitely many homoclinic solutions for a class of discrete Schrödinger equations in infinitemdimensional lattices with nonlinearities being superlinear at infinity by using variational methods. Our result extends some existing results in the literature.

MSC: 35Q51; 35Q55; 39A12; 39A70

Keywords: discrete nonlinear Schrödinger equations; variational methods; superlinear; homoclinic solutions

1 Introduction and main results

The discrete nonlinear Schrödinger equation is one of the most important discrete mod-els, which plays an important role in many ﬁelds; for example, in biomolecular chains [], nonlinear optics [], Bose-Einstein condensates [], and so on. In recent decades, a lot of results have been achieved in the study of homoclinic solutions for periodic discrete nonlinear Schrödinger equations; see [–], and so on. But we notice that there are only a few results of non-periodic discrete nonlinear Schrödinger equations, such as [–]. The authors of [, , , ] studied the case in inﬁnite one dimensional lattices (i.e.,

n∈Z), but the authors of [, , , , ] studied the case in inﬁnitemdimensional lattices (i.e.,n∈Zm_).

Inspired by the above results, we will study homoclinic solutions of the following non-periodic discrete nonlinear equation in inﬁnitemdimensional lattices by more general conditions than some existing results:

–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–)

is the discrete Laplace operator inmdimensional space,ω∈R,V={vn}n∈Zm, and{u_n}_n∈Zm

are sequences of real numbers, and the nonlinearitiesfnsatisfy the condition:

fn

eiωs=eiωfn(s), ∀ω∈R,∀(n,s)∈Zm×R.

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As usual, homoclinic solutions of equation (.) satisfy the following boundary condition:

lim

|n|=|n|+|n|+···+|nm|→∞

un= . (.)

Here we are interested in the existence of inﬁnitely many nontrivial homoclinic solutions for (.) (‘uis nontrivial’ meansun≡). The problem (.) comes from the study of

stand-ing waves for the discrete nonlinear Schrödstand-inger equation

iψ˙n= –ψn+vnψn–fn(ψn), n∈Zm. (.)

By the deﬁnition of standing waves (ψn=une–iωt with (.)), we see that (.) becomes

(.). Therefore, the problem of the existence of standing waves of (.) has been reduced to that on the existence of homoclinic solutions of (.).

In order to overcome the diﬃculties caused by the unboundedness ofZm_{and the lack}

of periodic conditions, we make some suitable assumptions and get the following result.

Theorem . The problem (.) has inﬁnitely many nontrivial homoclinic solutions if fn(–s) = –fn(s)for all(n,s)∈Zm×R and the following conditions hold:

(V) V={vn}n∈Zmis bounded from below and satisﬁes

lim

|n|→+∞vn= +∞. (.)

(F) fn∈C(R,R),fn(s) =o(s)ass→,and there exista> andν> such that

fn(s)≤a

 +|s|ν–, ∀(n,s)∈Zm×R.

(F) lim|s|→+∞F_|s|n(s)= +∞,∀n∈Zm,whereFn(s) :=

s

fn(t)dt, (n,s)∈Zm×R.

(F) There exist two positive constantsband>max{,ν– }such that

lim inf

|s|→+∞

fn(s)s– Fn(s)

|s| ≥b, ∀n∈Z

m_.

(F) _fn(s)s>Fn(s)ifs= ,Fn(s)≥,∀(n,s)∈Zm×R,and

lim inf

|s|→

fn(s)s– Fn(s)

|s|ι ≥a for somea> andι∈[,ν],∀n∈Z

m_.

To explain the rationality of the assumptions for the nonlinear terms fn, we give the

following example. It is easy to check that the functions given in the following example satisfy our assumptions.

Example . Let

Fn(s) =an

|s|p+ (p– )|s|p–εsin|s|ε/ε, s∈R,

wherean≥C>  for alln∈Zm,p> , and  <ε<p– . Note that

fn(s)s– Fn(s) = (p– )an

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Remark . Our result extends some results [, , , , ] in inﬁnitemdimensional lattices.

() The results [, , , ] are all about the positive deﬁnite case (ω<infσ(–+V)), but the temporal frequencyω∈Rin our paper.

() The authors of [, , , ] all used the conditions(V),(F), and(F). Besides, the authors of [, ] also used the following monotony condition:

fn(s)

s is increasing fors> and decreasing fors< . (.)

The authors [, ] also used the following Ambrosetti-Rabinowitz superlinear condition: there existsν> such that

 <νFn(s)≤fn(s)s, ∀s∈R\{}. (.)

But we use local conditions(F)and(F)to replace the conditions (.) and (.). The functions of Example . satisfy our conditions(F)-(F), but they do not satisfy (.) and (.), which shows that our conditions are weaker than the above

conditions.

() The results in [] also rely on the monotony condition (.).

In Section , we establish the variational framework of (.) and give some preliminary lemmas. In Section , we give the detailed proof of our main result.

2 Preliminary lemmas Let

lp≡lpZm:=

u={un}:n∈Zm,un∈R,ulp=

n∈Zm

|un|p

/p

<∞

,

p∈[, +∞),

be real sequence spaces. Clearly, the following elementary embedding relations hold:

lp⊂lq, ulq≤ u_lp, ≤p≤q≤ ∞, whereu_l∞:=max

n∈Zm|un|.

LetL:= –+V be deﬁned byLun:= –un+vnunforu∈l. LetE be the form domain

ofL,i.e.,E:=D(L/_{) (the domain of}_L/_{). Under our assumptions, the operator}_L_{is an} unbounded self-adjoint operator inl_{. Since the operator –}_{is bounded in}_l_{, it is easy to} see thatE={u∈l_:_V/_u_∈_l_}_{, where}_V/_u_{is deﬁned by}_V/_u

n:=v/n unforu∈l. We

deﬁne, respectively, onEthe inner product and norm by

(u,v)E:= (u,v)l+

L/u,L/v_l and uE= (u,u)/E ,

where (u,v)l is the inner product inl. ThenEis a Hilbert space.

Lemma .([]) If(.)holds,then we have:

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() The spectrumσ(L)is discrete and consists of simple eigenvalues accumulating to +∞.

By Lemma .(), we can assume that

λ–ω<λ–ω<· · ·<λk–ω<· · · →+∞

are all eigenvalues of L–ωandekis the associated normalized eigenfunction with the

eigenvalueλk–ωfor eachk,i.e., (L–ω)ek= (λk–ω)ekandekl= ,k= , , . . . . Moreover, {ek:k= , , . . .}is an orthonormal basis ofl. Let(D) denote the numberiwithi∈D. Let

k:=

{i:λi–ω< }

, k:=

{i:λi–ω= }

, k:=k+k, (.)

and

E–:=span{e, . . . ,ek}, E

_:=_span_{_e

k+, . . . ,ek}, E

+_:=_span_{_e

k+, . . .},

where the closure is taken with respect to the norm · E. Then one has the orthogonal

decomposition

E=E–⊕E⊕E+

with respect to the inner product (·,·)E. Now, we introduce, respectively, onEthe following

inner product and norm:

(u,v) :=u,v_l+

L_u_,_L_v

l, u= (u,u)  _,

whereu,v∈E=E–⊕E⊕E+withu=u–+u+u+andv=v–+v+v+. Clearly, the norms · and · Eare equivalent, and the decompositionE=E–⊕E⊕E+is also orthogonal

with respect to both inner products (·,·) and (·,·)_l.

From the above arguments, we consider the functionalonEdeﬁned by

(u) =  

(L–ω)u,u_l–

n∈Zm

Fn(un)

= u

+ –

u –

–I(u),

whereI(u) :=_n∈ZmFn(un). Under our assumptions,I,∈C(E,R), and the derivatives

are given by

(u),v=u+,v+–u–,v––I(u),v, I(u),v=

n∈Zm

fn(un)vn,

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Lemma .([]) Let E=∞_j_=Xj (dimXj<∞,∀j∈N)be a Banach space with the norm in the Appendix, we see that there is a constant>  such that

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(F) implies that there areR,R>  such that

Fn(s)≥R|s|, ∀(n,s)∈Zm×Rwith|s| ≥R. (.)

For anyu∈Hwithu ≥R/, we have

|un| ≥R, ∀n∈u. (.)

Note thatFn(s)≥ for all (n,s)∈Zm×R, it follows from (.)-(.) and the deﬁnitions of B(u) anduthat, for anyu∈Hwithu ≥R/,

B(u) =  u

– +

n∈Zm

Fn(un)

≥

n∈u

Fn(un)

≥

n∈u

R|un|

≥Ru·(u)≥Ru.

It implies

B(u)→ ∞ asu → ∞onE–⊕E,

which is due toE–_⊕_E_{being of ﬁnite dimension. It follows from the fact}_E₌_E–_⊕_E_⊕_E+ and the deﬁnitions ofAandBthat we have

A(u)→ ∞ or B(u)→ ∞ asu → ∞,∀u∈E.

The proof is completed.

Lemma . If the assumptions in Theorem.are satisﬁed,then(F)in Lemma.holds.

Proof (a) Note that (F) implies that for anyε>  there existsCεsuch that

Fn(s)≤ε|s|+Cε|s|ν, ∀(n,s)∈Zm×R.

It follows from the deﬁnition ofλthat

λ(u)≥  u

_{– }

n∈Zm

Fn(un)

≥  u

_{– }

n∈Zm

εun+Cεun

ν

, ∀(λ,u)∈[, ]×E+. (.)

Let

l(k) := sup

u∈Zk\{}

u_l

u , lν(k) :=u∈Zsupk\{}

ulν

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Note that

l(k)→, lν(k)→ ask→ ∞, (.)

which will be proved in the appendix. Obviously,Zk⊂E+for allk≥k+  (k+  is deﬁned above Lemma .), thus it follows from (.)-(.) that for anyk≥k+  we have

λ(u)≥  u

_{– }_εl

(k)u– Cεlνν(k)u ν

, ∀(λ,u)∈[, ]×Zk. (.)

Let

ρk:=

 – εl_(k)Cεlνν(k)



–ν_. _(.)

By (.), there exists a large enoughk>k+  such that

 < εl_(k) < , ∀k>k. (.)

By (.), (.), (.), andν> , we have

ρk→ ∞ ask→ ∞. (.)

By (.)-(.), we have

αk(λ) := inf u∈Zk,u=ρk

λ(u)≥ρ_k/ > , ∀k≥k.

(b) Note thatYkis of ﬁnite dimension, thus (.) implies that for anyk∈Nthere exists

a constantk>  such that

n∈Zm:|un| ≥ku

≥, ∀u∈Yk\{}. (.)

By (F), for anyk∈N, there exists a constantSk>  such that

Fn(s)≥

|s|



k

, ∀(n,s)∈Zm×Rwith|s| ≥Sk. (.)

For anyk∈Nandu∈Ykwithu ≥Sk/k, by (.), (.), and the factFn(s)≥, we have

λ(u)≤  u

+ –

n∈Zm

Fn(un)

≤  u

_–

n∈{n∈Zm_:_|u n|≥ku}

|un| 

k

≤  u

_–ku 

k

·n∈Zm:|un| ≥u

≤  u

_–_u_{= –} u

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Now for anyk∈N, if we choose

rk>max{ρk,Sk/k},

then (.) implies that

βk(λ) := max u∈Yk,u=rk

λ(u)≤–r_k/ < , ∀k∈N.

Therefore, the proof is ﬁnished.

3 Proof of the main result

Proof of Theorem. It is easy to check that (F) of Lemma . holds. Besides, (F) and (F) hold for allk≥kby Lemmas . and .. Thus Lemma . implies that for anyk≥kand a.e.λ∈[, ] there exists a sequence{uk

i(λ)}∞i=⊂Esuch that

sup

i

_uk

i(λ)<∞, λ

uk_i(λ)=  and λ

uk_i(λ)→ζk(λ) asi→ ∞, (.)

where

ζk(λ) := inf

γ∈k

max

u∈Bk

λ

γ(u), ∀λ∈[, ],

withBk:={u∈Yk:u ≤rk}andk:={γ ∈C(Bk,E)|γ is odd,γ|∂Bk=id}. Furthermore, it

follows from the proof of Lemma . that

ζk(λ)∈[αk,ζk], ∀k≥k, (.)

where ζ_k:=maxu∈Bk(u) andαk:=ρ



k/→ ∞ask→ ∞by (.). By (.), for each k≥k, there existλj→ asj→ ∞and{uki(λj)}∞i=⊂Esuch that

sup

i

uk_i(λj)<∞, λj

uk_i(λj)

=  and λj

uk_i(λj)

→ζk(λj)

asi→ ∞. (.)

Claim  {uk_i(λj)}∞i=in(.)has a strong convergent subsequence.

Proof Note thatsup_iuk

i(λj)<∞for eachk≥k, without loss of generality, we may as-sume

uk_i(λj)

–

→uk_j–, uk_i(λj)



→uk_j and uk_i(λj)

+

uk_j+

asi→ ∞,∀j∈N,

(.)

for someuk

j = (ukj)–+ (ujk)+ (ukj)+∈E=E–+E+E+sincedim(E–⊕E) <∞. By virtue

of the Riesz representation theorem,_λ

j:E→E

∗_and_I_:_E_→_E∗_{can be viewed as}

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E→EandI:E→E, respectively, whereE∗is the dual space ofE. Note that (.) implies that for eachk≥k

 =_λ_juk_i(λj)

=uk_i(λj)

+ –λj

uk_i(λj)

–

+Iuk_i(λj)

, ∀i,j∈N,

that is,

uk_i(λj)

+ =λj

uk_i(λj)

–

+Iuk_i(λj)

, ∀i,j∈N. (.)

By the standard argument (see [, ]), we knowI:E→E∗ is compact. Therefore,I:

E→E is also compact. It follows from (.) and (.) that the right-hand side of (.) converges strongly inE. Combining this with (.), we have

lim

i→∞u k

i(λj) =ukj ∈E, ∀j∈Nandk≥k. (.)

So Claim  is true.

By (.), (.), and (.), we have

_λ

j

uk_j=  and λj

uk_j∈[αk,ζk], ∀j∈Nandk≥k. (.)

In fact, we can see{uk_j}∞_j_=is bounded inE, which will be proved in the appendix. Besides, by a similar proof to Claim , we can also see that{uk

j}∞j= possesses a strong convergent subsequence inEfor allk≥k. Without loss of generality, we may assume

uk_j →uk asj→ ∞,∀k≥k.

For eachk≥k, by (.), the limitukis just a critical point of=with(uk)∈[αk,ζk].

Sinceαk→ ∞ask→ ∞in (.), we get inﬁnitely many nontrivial critical points of.

Therefore, we see that problem (.) possesses inﬁnitely many nontrivial homoclinic so-lutions. The proof of Theorem . is completed.

Appendix

Fact  The result(.)holds.

Proof If not, for anyj∈N, there existsuj_∈_H_\{}_{such that}

n∈Zm:u_nj≥uj/j= .

Letvj_:= uj

uj∈H, thenvj=  and

n∈Zm:vj_n≥/j= , ∀j∈N. (A.)

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equivalent, thus by Lemma .(), we have

vj–v_l=

n∈Zm

vj_n–vn



→ asj→ ∞. (A.)

The fact thatv=  impliesvl∞=maxn∈Zm|v_n|> . By the deﬁnition of the norm·_l∞,

there exists a constantδ>  such that

n∈Zm:|vn| ≥δ

≥. (A.)

For anyj∈N, let

j:=

n∈Zm:vj_n< /j and c_j:=Zm\j=

n∈Zm:vj_n≥/j.

Set:={n∈Zm:|vn| ≥δ}. Then forjlarge enough, by (A.) and (A.), we have

(j∩)≥() –

c_j≥ –  = .

It follows from the deﬁnitions ofjandthat forjlarge enough we have

n∈Zm

vj_n–vn≥

n∈j∩

vj_n–vn

≥

n∈j∩ |vn|

|vn|– vjn

≥δ(δ– /j)·(j∩)

≥δ_/ > .

This is in contradiction to (A.). Therefore, (.) holds.

Fact  The result(.)holds.

Proof It is clear that  <lν(k+ )≤lν(k), so thatlν(k)→l≥ ask→ ∞. For everyk≥, there existsuk_∈_Z

ksuch thatuk=  anduklν>l_ν(k)/. By the deﬁnition ofZ_k,uk

inE, thenuk_→_{ in}_lν_{dues to Lemma .(). Therefore, we have}_l_{= , that is,}_l

ν(k)→. Similarly,l(k)→. Therefore, (.) holds.

Fact  {uk

j}∞j=is bounded in E.

Proof Note that (F) implies that there exists a constantL>  such that



fn(s)s–Fn(s)≥

b

|s|

_, _∀₍_n_,_s₎_∈_Zm_×_R_with_|_s_{| ≥}_L

. (A.)

For notational simplicity, we will set

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Then by (A.) and the deﬁnitions ofjandχj, we have

( –χj)uj_l∞≤L and χjujl=

n∈j

|uj,n|≤D, ∀j∈N.

It follows from the equivalence of any two norms on the ﬁnite-dimensional spaceE⊕E–

and Hölder’s inequality that

Therefore, by (.), (A.)-(A.), and the Sobolev embedding theorem we have

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≤c+cεuj+cLν–ι

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

Research supported by National Natural Science Foundation of China (No. 11401011).

We would like to thank the editors and referees for their valuable comments which have led to an improvement of the presentation of this paper.

Received: 19 September 2016 Accepted: 17 October 2016 References

1. Kopidakis, G, Aubry, S, Tsironis, GP: Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett.87, 165501 (2001)

2. Christodoulides, DN, Lederer, F, Silberberg, Y: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature424, 817-823 (2003)

3. Livi, R, Franzosi, R, Oppo, GL: Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett.97, 060401 (2006)

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. Pankov, A: Gap solitons in periodic discrete nonlinear equations. Nonlinearity19, 27-40 (2006)

8. Pankov, A: Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst.19, 419-430 (2007)

9. Pankov, A, Rothos, V: Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. Proc. R. Soc. A464, 3219-3236 (2008)

10. Pankov, A: Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. J. Math. Anal. Appl.371, 254-265 (2010)

11. Yang, M, Chen, W, Ding, Y: Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math.

110, 1475-1488 (2010)

12. Zhou, Z, Yu, J, Chen, Y: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity23, 1727-1740 (2010)

13. Shi, H, Zhang, H: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl.

361, 411-419 (2010)

14. Shi, H: Gap solitons in periodic discrete Schrödinger equations with nonlinearity. Acta Appl. Math.109, 1065-1075 (2010)

(14)

16. 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)

17. Pankov, A: Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities. Appl. Anal.92, 308-317 (2013)

18. Sun, G: On standing wave solutions for discrete nonlinear Schrödinger equations. Abstr. Appl. Anal.2013, 436919 (2013)

19. Zhang, G, Pankov, A: Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. Commun. Math. Anal.5, 38-49 (2008)

20. Zhang, G, Liu, F: Existence of breather solutions of the DNLS equations with unbounded potentials. Nonlinear Anal.

71, 786-792 (2009)

21. Zhang, G: Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials. J. Math. Phys.50, 013505 (2009)

22. Zhang, G, Pankov, A: Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, II. Appl. Anal.89, 1541-1557 (2010)

23. Zhou, Z, Ma, D: Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials. Sci. China Math.58, 781-790 (2015)

24. Zou, W: Variant fountain theorems and their applications. Manuscr. Math.104, 343-358 (2001)

25. Benci, V, Rabinowitz, PH: Critical point theorems for indeﬁnite functionals. Invent. Math.52, 241-273 (1979) 26. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Diﬀerential Equations. CBMS Reg.