On sequences of large homoclinic solutions for a difference equations on the integers

R E S E A R C H

Open Access

On sequences of large homoclinic

solutions for a difference equations on the

integers

Robert Stegli ´nski

*_{Correspondence:}

robert.steglinski@p.lodz.pl Institute of Mathematics, Technical University of Lodz, Wolczanska 215, Lodz, 90-924, Poland

Abstract

In this paper, we determine a concrete interval of positive parameters

λ

–

φ

(

)

φ

(

)

λ

(

)

where the nonlinear termf:Z×R→Rhas an appropriate behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.

MSC: 39A10; 47J30; 35B38

Keywords: difference equations; discretep-Laplacian; variational methods; infinitely many solutions

1 Introduction

In the present paper we deal with the following nonlinear second-order difference equa-tion:

–φp(u(k– )) +a(k)φp(u(k)) =λf(k,u(k)) for allk∈Z,

u(k)→ as|k| → ∞. ()

Herep>  is a real number,λis a positive real parameter,φp(t) =|t|p–t for allt∈R,

a:Z→Ris a positive weight function, while f :Z×R→Ris a continuous function. Moreover, the forward difference operator is defined asu(k– ) =u(k) –u(k– ). We say that a solutionu={u(k)}of () is homoclinic iflim_|k|→∞u(k) = .

Difference equations represent the discrete counterpart of ordinary differential equa-tions and are usually studies in connection with numerical analysis. We may regard () as being a discrete analog of the following second-order differential equation:

–φp

x(t)+a(t)φp

x(t)=ft,x(t), t∈R.

However, the relations between discretization and its continuous counterpart are not as direct as it may seem; see for example []. The casep=  in () has been motivated in part

by searching standing waves for the nonlinear Schrodinger equation

iψ˙k+ψk–νkψk+f(k,ψk) = , k∈Z

(for details see [, ]).

Most of the classical methods used in the case of differential equations lead to exis-tence results for the solutions of a difference equation. Variational methods for difference equations consist in seeking solutions as critical points for a suitable associated energy functional defined on a convenient Banach space. In the first approaches to the issue, the variational methods are applied to boundary value problems on bounded discrete in-tervals, which leads to the study of an energy functional defined on a finite-dimensional Banach space (see [–]). In the case of difference equations on unbounded discrete in-tervals (typically, on the whole set of integersZ) solutions are sought in a subspace of the sequence spacelp_{which is still infinite-dimensional but compactly embedded intol}p_(see

[–]).

Our idea here is to investigate the existence of infinitely many solutions to problem () by using a critical point theorem obtained in []; see Theorem . This theorem was used in a continuous case in [] and in discrete but finite-dimensional analog of problem () in [–]. In analogy with the cited papers, in our approach we do not require any symmetry hypothesis.

A special case of our contributions reads as follows. Fora:Z→Rand the continuous mappingf:Z×R→Rdefine the following conditions:

(A) a(k)≥α> for allk∈Z,a(k)→+∞as|k| →+∞; (F) limt→f_t(pk–,t)= uniformly for allk∈Z;

(F) lim inft→+∞

k∈Zmax_|ξ|≤tF(k,ξ)

tp = ; (F) lim sup(k,t)→(+∞,+∞)(+Fa((kk,))|t)t|p= +∞; (F) supk∈Z(lim supt→+∞(+Fa((kk,))|t)t|p) = +∞,

whereF(k,t) is the primitive function off(k,t), that is,F(k,t) =_tf(k,s)dsfor everyt∈R

andk∈Z. The solutions are found in the normed space (X, · ), whereX={u:Z→R:

k∈Za(k)|u(k)|p<∞}andu= [

k∈Za(k)|u(k)|p]  p_.

Theorem  Assume that(A), (F),and(F)are satisfied.Moreover,assume that either(F) or(F)is satisfied.Then for eachλ> ,the problem()admits a sequence of solutions in X whose norms tend to infinity.

The plan of the paper is the following: Section  is devoted to our abstract framework, while Section  is dedicated to the main results. Finally, we give two examples which illus-trate the independence of conditions (F) and (F) and a third example which corresponds

to the general Theorem .

2 Abstract framework

Theorem  Let(E, · )be a reflexive real Banach space,let ,:E→Rbe two contin-uously differentiable functionals with coercive,i.e.limu→∞ (u) = +∞,and a sequen-tially weakly lower semicontinuous functional anda sequentially weakly upper semicon-tinuous functional.For every r>infE ,let us put

ϕ(r) := inf

u∈ –_((–∞,r))

(sup_v∈ –_((–∞,r))(v)) –(u)

r– (u)

and

γ :=lim inf

r→+∞ ϕ(r).

Let Jλ:= (u) –λ(u)for all u∈E.Ifγ < +∞then,for eachλ∈(,_γ),the following alter-native holds:

either

(a) Jλpossesses a global minimum,

or

(b) there is a sequence{un}of critical points(local minima)ofJλsuch that

lim_n→+∞ (un) = +∞.

We begin by defining some Banach spaces. For all ≤p< +∞, we denotepthe set of all functionsu:Z→Rsuch that

up_p=

k∈Z

u(k)p< +∞.

Moreover, we denote by∞the set of all functionsu:Z→Rsuch that

u∞=sup

k∈Z

u(k)< +∞.

We set

X= u:Z→R:

k∈Z

a(k)u(k)p<∞

and

u=

k∈Z

a(k)u(k)p

 p

.

Clearly we have

u∞≤ up≤α– 

p_{u for allu}∈_{X. ()}

As is shown in [], Proposition , (X,·) is a reflexive Banach space and the embedding

X→lpis compact. Let

(u) :=

p

k∈Z

and

(u) :=

k∈Z

Fk,u(k) for allu∈lp,

whereF(k,s) =_sf(k,t)dtfors∈Randk∈Z. LetJλ:X→Rbe the functional associated to problem () defined by

Jλ(u) = (u) –λ(u).

Proposition  Assume that(A)and(F)are satisfied.Then

(a) ∈C_(X);

(b) ∈C_(lp_)and∈C_(X);

(c) Jλ∈C_{(X)and for allλ> every critical pointu∈XofJλis a homoclinic solution of} problem().

For a proof see Propositions , , and  in [].

Proposition  Assume that(A)and(F)are satisfied.Then is coercive and sequentially weakly lower semicontinuous functional andis sequentially weakly upper semicontinu-ous functional.

Proof We have

(u)≥

pu

p_→+∞,

asu →+∞, and so is coercive. Now, letumuweakly inX. By the compactness

of the embeddingX→lp_{, we may assume that u}

m→uin lp. The rest of the

propo-sition follows from the fact that the norm · inX is sequentially weakly lower semi-continuous functional, the functional is continuous onlp, and the functionallp u→



p

k∈Z|u(k– )|p∈Ris continuous onlp.

Proposition  Let{um}be a sequence in X such that (um)→+∞.Thenum →+∞.

Proof By the classical Minkowski inequality

_n

i=

|xi+yi|p 

p ≤

_n

i=

|xi|p 

p

+

_n

i=

|yi|p 

p

for alln∈N,x, . . . ,xn,y, . . . ,yn∈R, we have

|k|≤h

u(k– )p

 p

≤

|k|≤h

_u(k– )p

 p

+

|k|≤h _u(k)p

 p

for allh∈Nandu∈X. Lettingh→+∞we obtain

k∈Z

u(k– )p

 p

From this and inequality () we conclude

(u) = 

p

k∈Z

u(k– )p+a(k)u(k)p

≤ 

p

up_p+up≤ +α pα u

p_,

which proves the proposition.

3 Main theorem

Now we will formulate and prove a stronger form of Theorem . Let

A:=lim inf

t→+∞

k∈Zmax|ξ|≤tF(k,ξ)

tp ,

B±,±:= lim sup (k,t)→(±∞,±∞)

F(k,t) ( +a(k))|t|p,

and

B±:=sup

k∈Z

lim sup

t→±∞

F(k,t) ( +a(k))|t|p

.

SetB:=max{B±,±,B±}. For convenience we put + = +∞and_+∞ = .

Theorem  Assume that(A)and(F)are satisfied and assume that the following inequal-ity holds:A<α·B.Then,for eachλ∈(

Bp,

α

Ap),problem()admits a sequence of solutions

in X whose norms tend to infinity.

Proof It is clear thatA≥. Putλ∈(_Bp ,_Apα) and put ,,Jλas in the previous section. Our aim is to apply Theorem  to the functionJλ. By Lemma , the functional is the locally Lipschitz, coercive and sequentially weakly lower semicontinuous functional and

is the locally Lipschitz, sequentially weakly upper semicontinuous functional. We will show thatγ < +∞. Let{cm} ⊂(, +∞) be a sequence such thatlimm→∞cm= +∞and

lim

m→+∞

k∈Zmax|ξ|≤cmF(k,ξ)

cpm

=A.

Set

rm:=

α pc

p m

for everym∈N. Then, ifv∈Xand (v) <rm, one has v∞≤α–



p_v ≤_α–p_{p (v)}  p _<c

m,

which gives

–_(–∞,r

m)

⊂v∈X:v∞≤cm

From this and () =() =  we have

Now, our conclusion follows provided thatJλdoes not possess a global minimum.

Then there exists a sequence of real numbers{bm}withbm≥ such thatlimm→+∞bm=

+∞and

F(k,bm) >M

 +a(k)

bp_m

for allm∈N. Thus, take inXa sequence{sm}such that, for everym∈N,sm(k) =bm, and

sm(k) =  for everyk∈Z\{k}. Then one has

Jλ(sm) =



p

k∈Z

sm(k– ) p

+a(k)sm(k) p

–λ

k∈Z

Fk,sm(k)

< 

pb

p m+



pa(k)b

p m–λM

 +a(k)

bp_m

=



p–λM

 +a(k)

bp_m,

which giveslim_m→+∞Jλ(sm) = –∞.

Next, assume thatB< +∞. Sinceλ>_Bp , we can fixε>  such thatε<B–_λ_p. Therefore, there exists an indexk∈Zsuch that

lim sup

t→+∞

F(k,t)

( +a(k))|t|p

>B–ε,

and taking{bm}, a sequence of real numbers withbm≥ such thatlimm→+∞bm= +∞and

F(k,bm) > (B–ε)

 +a(k)

bp_m

for allm∈N, choosing{sm}inXas above, one has

Jλ(sm) <



p–λ(B–ε)

 +a(k)

bp_m.

So, also in this case,limm→+∞Jλ(sm) = –∞. Finally, the above facts mean thatJλdoes not

possesses a global minimum. Hence, by Theorem , we obtain a sequence{um}of critical

points (local minima) ofJλsuch thatlimm→+∞ (um) = +∞. Proposition  now implies

lim_m→+∞um= +∞. The proof is complete.

Remark  Theorem  follows from Theorem  since the condition (F) means thatB+,+=

+∞and the condition (F) means thatB+= +∞, soB= +∞. Moreover, (F) impliesA= .

Hence, we obtain the interval of all positive parameters, for which the treated problem admits infinitely many solutions whose norms tend to infinity.

Remark  If we replace the condition (F) by

(F+

) limt→+ f(k,t)

tp– = uniformly for allk∈Z

in Theorem , then we can obtain the sequence of positive solutions of the problem (). It can be proved by using Lemmas  and  in [].

4 Examples

Example  Let{a(k)}be a sequence of positive numbers. Putb= . Let{bm},{cm},{hm}

be sequences defined recursively by

⎧ every positive integerkletf(k,·) :R→Rbe any nonnegative continuous function such thatf(k,t) =  fort∈R\(bk,ck) and

be sequences defined recursively by

It is easy to see thatbm<cm<bm+ for everym∈Nandlimm→+∞bm=limm→+∞cm= Letα,β be any two strictly positive real numbers. We show that there is a continuous functionf :Z×R→R, such that the numbersAandB, as defined above Theorem , are equal toαandβ, respectively. We can follow one of two strategies given by Example  and Example . We choose the first one.

It is easy to see thatbm<cm<bm+for everym≥mandlimm→+∞bm=limm→+∞cm=

+∞. For every integerk<mletf(k,·) :R→Rbe identically the zero function. For

ev-ery integerk≥mletf(k,·) :R→Rbe any nonnegative continuous function such that f(k,t) =  fort∈R\(bk,ck) and

ck

bkf(k,t)dt=hk. The condition (F) is now obviously

sat-isfied.

SetF(k,t) :=_tf(k,s)dsfor everyt∈Randk∈Z. Since

bpm

cpm ·

k∈Zmax|ξ|≤bmF(k,ξ)

bpm

=

k∈Zmax|ξ|≤cm–F(k,ξ)

cpm

≤ inf

t∈(cm–,cm)

k∈Zmax|ξ|≤tF(k,ξ)

tp

≤

k∈Zmax|ξ|≤bmF(k,ξ)

bpm

is satisfied for allm≥m, we have

lim inf

t→+∞

k∈Zmax|ξ|≤tF(k,ξ)

tp =_mlim_→+∞

k∈Zmax|ξ|≤bmF(k,ξ)

bpm

and since

F(k,ck)

( +a(k))cp_k ≤supt∈R

F(k,t) ( +a(k))|t|p ≤

hk

( +a(k))bp_k =

F(k,ck)

( +a(k))cp_k · cp_k bp_k

is satisfied for allk≥m, we have

lim sup (k,t)→(+∞,+∞)

F(k,t)

( +a(k))|t|p =_k→+∞lim

F(k,ck)

( +a(k))cp_k.

In the same way as has been done in Example , we conclude that

lim inf

t→+∞

k∈Zmax|ξ|≤tF(k,ξ)

tp =α

and

lim sup (k,t)→(+∞,+∞)

F(k,t) ( +a(k))|t|p =β.

Moreover,

sup

k∈Z

lim sup

t→+∞

F(k,t) ( +a(k))|t|p

= .

From this we obtainA=αandB=β.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

Acknowledgements

The research was supported by Lodz University of Technology. The author would like to thank the referees for their valuable suggestions and comments.

Received: 18 November 2015 Accepted: 25 January 2016

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