Volume 2010, Article ID 907453,21pages doi:10.1155/2010/907453
Research Article
Existence of Multiple Solutions of a Second-Order
Difference Boundary Value Problem
Bo Zheng and Huafeng Xiao
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
Correspondence should be addressed to Bo Zheng,[email protected]
Received 3 July 2009; Revised 5 February 2010; Accepted 7 March 2010
Academic Editor: Martin Bohner
Copyrightq2010 B. Zheng and H. Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper studies the existence of multiple solutions of the second-order difference boundary value problemΔ2un−1 Vun 0,n∈ Z1, T,u0 0 uT1. By applying Morse
theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalueλk k≥2of linear
difference problemΔ2un−1 λun 0,n∈Z1, T,u0 0uT1near infinity and the trivial solution of the first equation is a local minimizer under some assumptions onV.
1. Introduction
LetR,N, andZbe the sets of real numbers, natural numbers, and integers, respectively. For anya, b∈Z,a≤b, defineZa, b {a, a1, . . . , b}.
Consider the second-order difference boundary value problemBVP
Δ2un−1 Vun 0, n∈Z1, T,
u0 0uT1,
1.1
where V ∈ C2R,Rand Δdenotes the forward difference operator defined byΔun
un1−un,Δ2un ΔΔun.
By a solution u of the BVP 1.1, we mean a real sequence {un}Tn01 u0, u1, . . . , uT1satisfying the BVP1.1. Foru{un}Tn10withu0 0uT1,
if for alln ∈ Z1, T,un < 0. The aim of this paper is to obtain the existence of multiple solutions of the BVP1.1and analyse the sign of solutions.
Recently, a few authors applied the minimax methods to examine the difference boundary value problems. For example, in1 , Agarwal et al. employed the Mountain Pass Lemma to study the following BVP:
Δ2un−1 fn, un 0, n∈Z1, T,
u0 0uT1
1.2
and obtained the existence of multiple positive solutions, wherefmay be singular atu0. In 2 , Jiang and Zhou employed the Mountain Pass Lemma together with strongly monotone operator principle, to study the following difference BVP:
Δ2un−1 fn, un 0, n∈Z1, T,
u0 0 ΔuT
1.3
and obtained existence and uniqueness results, wheref :Z1, T×R → Ris continuous. In 3 , Cai and Yu employed the Linking Theorem and the Mountain Pass Lemma to study the following difference BVP:
ΔpnΔun−1δqnuδn fn, un, n∈Z1, T,
Δu0 A, uT1 B
1.4
and obtained the existence of multiple solutions, where δ > 0 is the ratio of odd positive integers,{pn}nT11and{qn}Tn1are real sequences,pn/0 for alln∈Z1, T 1, andA,B
are two given constants,f :Z1, T×R → Ris continuous.
Although applications of the minimax methods in the field of the difference BVP have attracted some scholarly attention in the recent years, efforts in applying Morse theory to the difference BVP are scarce. The main purpose of this paper is to develop a new approach to the BVP1.1by using Morse theory. To this end, we first consider the following linear difference eigenvalue problem:
Δ2un−1 λun 0, n∈Z1, T,
u0 0uT1. 1.5
On the above eigenvalue problem, the following results hold; see4 .
Proposition 1.1. The eigenvalues of 1.5are
λλl4 sin2 lπ
2T1, l1,2, . . . , T, 1.6
Remark 1.2. 1 The set of functions {φln, l 1,2, . . . , T} is orthogonal on Z1, T with
respect to the weight functionrn≡1, that is,
T
n1
φln, φjn
0 ∀l /j. 1.7
Moreover, for eachl∈Z1, T,Tn1sin2lπn/T1 T1/2.
2It is easy to see thatφ1is positive andφl changes sign for eachl ∈Z2, T, that is,
{n:φln>0}/∅and{n:φln<0}/∅.
For1.1, we assume that
V0 V0 0, 1.8
V∞: lim
|t| → ∞
Vt
t λk, 1.9
whereλkis an eigenvalue of1.5. Hence the BVP1.1has a trivial solutionu≡0. And we
say that BVP1.1is resonant at infinity if1.9holds. Let
W−spanφ1, φ2, . . . , φk−1 , W0span
φk , Wspan
φk1, φk2, . . . , φT .
1.10
LetGt t0GsdsVt−λk/2t2.By1.9we have
lim
|t| → ∞
Gt
t 0. 1.11
Assume that the following conditions onGthold.
G±Ifum → ∞such thatvm/um → 1, then there existδ >0 andM ∈ Nsuch
that
±T
n1
Gumn, vmn
≥δ, ∀m≥M, 1.12
whereumvmwm,vm∈W0,wm∈W :W⊕W−.
The main result of this paper is as follows.
hold. Then the BVP1.1has at least three nontrivial solutions, with one positive solution and one negative solution, in each of the following cases:
i(G) andk≥2;
ii(G−) andk≥3.
To the author’s best knowledge, only Bin et al. 5 deal with the existence and multiplicity of nontrivial periodic solutions for asymptotically linear resonant difference problem by the aid of Su6 . In5 ,Gsatisfies
Gz≤c
1|z|sc2, 1.13
lim
v → ∞vinf∈W0
1
v2sGz≥ 4β2
δT, 1.14
wherec1 >0,c2 >0,s∈0,1, βc1T1−s/2,δ >0. In5 , the authors obtained the existence
of one nontrivial periodic solution. Notice that1.13implies that1.11holds; however,G± is not covered by 1.14. In fact, conditions 1.13and 1.14 are borrowed from 6 . The conditions inTheorem 1.3coincide with the assumptions of Theorem 1 in7 . The aim of this paper is to develop a new approach to study the discrete systems by using Morse theory, minimax theorems, and some analysis technique. We wish to have some breakthrough points with the aid of the method of discretization.
The remaining part of this paper proceeds as follows. In the next section, we establish the variational framework of the BVP1.1and collect some results which will be used in the proof ofTheorem 1.3. InSection 3, we give the proof of Theorem 1.3. Finally, inSection 4, we give an example to illustrate our main result and summarize conclusions and future directions.
2. Variational Framework and Auxiliary Results
Let
Eu:u{un}nT01 withu0 0uT1∈R
. 2.1
Ecan be equipped with the norm · and the inner product·,·as follows:
u T
n0
|Δun|2 1/2
, ∀u∈E,
u, v
T
n0
Δun,Δvn, ∀u, v∈E,
where| · |denotes the Euclidean norm inRand·,·denotes the usual scalar product inR. It is easy to see thatE,·,·is a Hilbert space. Consider the functional defined onEby
Ju 1 2
T
n0
|Δun|2−
T
n1
Vun. 2.3
We claim that ifu∈ Eis a critical point ofJ, thenuis precisely a solution of the BVP1.1. Indeed, for everyu, v∈E, we have
Ju, v
T
n0
Δun,Δvn−
T
n1
Vun, vn−
T
n1
Δ2un−1 Vun, vn.
2.4
So, ifJu 0, then we have
T
n1
Δ2un−1 Vun, vn0. 2.5
Sincev∈Eis arbitrary, we obtain
Δ2un−1 Vun 0, n∈Z1, T. 2.6
Therefore, we reduce the problem of finding solutions of the BVP1.1 to that of seeking critical points of the functionalJinE.
According toProposition 1.1andRemark 1.2,Ecan be decomposed asEW−⊕W0⊕
W. For allu∈E, denoteuw0ww−withw0∈W0,w∈W, andw− ∈W−, then we
have the following Wirtinger type inequalities:
λ1
T
n1
un, un≤ u2 ≤λT T
n1
un, un, ∀u∈E, 2.7
λ1
T
n1
w−n, w−n≤w−2≤λk−1
T
n1
w−n, w−n, ∀w−∈W−, 2.8
λk1
T
n1
wn, wn≤ w2≤λT T
n1
wn, wn, ∀w ∈W, 2.9
see4 for details.
Now we collect some results on Morse theory and the minimax methods. LetEbe a real Hilbert space andJ∈C1E,R. Denote
Jc{u∈E:Ju≤c}, Kc
u∈E:Ju 0, Ju c 2.10
Definition 2.1. The functionalJsatisfies thePScondition if any sequence{um} ⊂Esuch that
{Jum}is bounded andJum → 0 asm → ∞has a convergent subsequence.
In8 , Cerami introduced a weak version of thePScondition as follows.
Definition 2.2. The functionalJsatisfies the Cerami conditionCconditionif any sequence {um} ⊂ E such that {Jum} is bounded and1 umJum → 0 as m → ∞ has a
convergent subsequence.
If J satisfies the PS condition or the C condition, then J satisfies the following deformation condition which is essential in Morse theorycf.9,10 .
Definition 2.3. The functionalJsatisfies theDccondition at the levelc∈Rif for any >0
and any neighborhoodNofKc, there are >0 and a continuous deformationη:0,1 ×E →
Esuch that
1η0, u ufor allu∈E;
2ηt, u ufor allu /∈J−1c−, c ;
3Jηt, uis nonincreasing intfor anyu∈E; 4η1, Jc\ N⊂Jc−.
Jsatisfies theDcondition ifJsatisfies theDccondition for allc∈R.
Letu0be an isolated critical point ofJwithJu0 c∈R, and letUbe a neighborhood
ofu0, the group
CqJ, u0:HqJc∩U, Jc∩U\ {u0}, q∈Z, 2.11
is called theqth critical group ofJ atu0, whereHqA, Bdenotes theqth singular relative
homology group of the pairA, Bover a fieldF, which is defined to be quotientHqA, B
ZqA, B/BqA, B, where ZqA, B is the qth singular relative closed chain group and
BqA, Bis theqth singular relative boundary chain group.
LetK {u∈E:Ju 0}. IfJKis bounded from below bya∈ RandJ satisfies theDccondition for allc≤a, then the group
CqJ,∞:HqE, Ja, q∈Z, 2.12
is called theqth critical group ofJat infinity11 .
Assume that #K < ∞andJ satisfies theDcondition. The Morse-type numbers of the pairE, Jaare defined by
MqMqE, Ja
u∈K
dimCqJ, u, 2.13
and the Betti numbers of the pairE, Jaare
By Morse theory12,13 , the following relations hold:
q
j0
−1q−jMj≥ q
j0
−1q−jβj, q∈Z,
∞
q0
Mq
∞
q0
βq.
2.15
Thus, ifCqJ,∞0, for somek∈Z, then there must exist a critical pointuofJwith
CqJ, u0, which can be rephrased as follows.
Proposition 2.4. LetEbe a real Hilbert space andJ ∈C2E,R. Assume that #K<∞and thatJ
satisfies theDcondition. If there exists someq ∈ Zsuch that CqJ,∞ 0, thenJmust have a
critical pointuwithCqJ, u0.
In order to prove our main result, we need the following result about the critical group onCqJ,∞.
Proposition 2.5. Let the functionalJ:E → Rbe of the form
Ju 1
2Au, uQu, 2.16
whereA:E → Eis a self-adjoint linear operator such that 0 is isolated inσA, the spectrum ofA. Assume thatQ∈C1E,Rsatisfies
lim
u → ∞
Qu
u 0. 2.17
DenoteV : kerA,W : V⊥ W ⊕W−, whereW (W−) is the subspace of Eon whichAis positive (negative) definite. Assume thatμ:dimW−,ν :dimV /0 are finite and thatJsatisfies theDcondition. Then
CqJ,∞∼δq,k±F, q∈Z, 2.18
provided thatJsatisfies the angle conditions at infinity.
AC∞±: there existM >0 andα∈0,1such that
±Ju, v ≥0 foruvw, u ≥M, w ≤αu, 2.19
wherekμ,k− μν,v∈V, andw∈W.
is a slight improvement of 11, Proposition 3.10 by Su and Zhao 7 . There are many other papers considering concrete problems by computing the critical groups at infinity with different methods, for example, see14–17.
We will use the Mountain Pass Lemmacf.12,18 in our proof.
LetBρdenote the open ball inEabout 0 of radiusrand let∂Bρdenote its boundary.
Theorem 2.7mountain pass lemma. LetEbe a real Banach space andJ ∈ C1E,Rsatisfying
the (PS) condition. SupposeJ0 0 and that
J1there are constantsρ >0, a >0 such thatJ|∂Bρ≥a >0, J2there is au0 ∈E\Bρsuch thatJu0≤0,
thenJpossesses a critical valuec≥a. Moreoverccan be characterized as
cinf
h∈Γssup∈0,1Jhs, 2.20
where
Γ {h∈C0,1 , E|h0 0, h1 u0}. 2.21
Definition 2.8mountain pass point. An isolated critical point uof J is called a mountain pass point, ifC1J, u0.
The following result is useful in computing the critical group of a mountain pass point; see13,19 for details.
Theorem 2.9. LetEbe a real Hilbert space. Suppose thatJ∈C2E,Rhas a mountain pass pointu, and thatJuis a Fredholm operator with finite Morse index, satisfying
Ju0≥0, 0∈σ
Ju0
⇒dim kerJu0
1, 2.22
then
CqJ, u0∼δq,1F, q∈Z. 2.23
3. Proof of
Theorem 1.3
We give the proof ofTheorem 1.3in this section. Firstly, we prove that the functionalJsatisfies theCconditionLemma 3.1and compute the critical groupCqJ,∞ Lemma 3.2. Then,
we employ the cut-offtechnique and the Mountain Pass Lemma to obtain two critical points u, u− of J and compute the critical groups CqJ, uand CqJ, u− Lemmas 3.3and 3.4.
Finally, we proveTheorem 1.3. Rewrite the functionalJas
Ju 1 2
T
n0
|Δun|2−λk 2
T
n1
|un|2−
T
n1
Lemma 3.1. Let 1.8 and 1.9 hold. If G satisfies (G±), then the functional J satisfies the C condition.
Proof. We only prove the case whereGholds. Let{um} ⊂Esuch that
Jum−→c∈R, 1umJum−→0 as m−→ ∞. 3.2
Then for allϕ∈E, we have
Jum, ϕum, ϕ −λk T
n1
umn, ϕn
−T
n1
Gumn, ϕn
. 3.3
Denoteum vmwm w−m with vm ∈ W0,wm ∈ W and w−m ∈ W−. SinceEis a
finite-dimensional Hilbert space, it suffices to show that{um}is bounded. Suppose that {um}is
unbounded. Passing to a subsequence we may assume thatum → ∞asm → ∞.
By1.11, for any >0, there existsb∈Rsuch that
Gt≤|t|b, ∀t∈R. 3.4
Letϕwmin3.3. Then by2.7,2.9, and3.4, we have
c1wm2 :
1− λk
λk1
wm2
≤ wm
2−
λk T
n1
wmn, wmn
T
n1
Gumn, wmn
Jum, wm
≤ wm T
n1
|umn|b|wmn|
≤ wm
λ1λk1
umwm
b√T
λk1
wm
:c2wmc3umwm,
3.5
where
c11− λk
λk1 >0, c21
b√T
λk1
, c3
λ1λk1
And since >0 is arbitrary, we have
wm
um −→
0 as m−→ ∞. 3.7
Similarly, letϕw−min3.3, by2.8and3.4we get
−c4wm−
2
:
1− λk λk−1
wm−
2
≥w−m
2−
λk T
n1
wm−n, w−mn
T
n1
Gumn, w−mn
Jum, w−m
≥ −w−m− T
n1
|umn|bwm−n
≥ −w−m−
λ1um
w−
m−
b√T λ1
w−
m
:−c5wm−−c6umw−m,
3.8
where
c4 λk
λk−1 −
1>0, c51b
√ T
λ1
, c6
λ1.
3.9
And hence we also have
wm−
um −→
0 as m−→ ∞. 3.10
By3.7and3.10, we have
wm
um −→0,
vm
um −→
1 as m−→ ∞. 3.11
ByG, there existδ >0 andM∈Nsuch that
T
n1
Gumn, vmn
This implies that
Jum, vm− T
n1
Gumn, vmn
≤ −δ, ∀m≥M, 3.13
and hence
Ju
mum ≥Jumvm ≥Jum, vm≥δ, ∀m≥M, 3.14
which is a contradiction to3.2. Thus{um}is bounded. The proof is complete.
Lemma 3.2. Let1.8and1.9hold. Then
1CqJ,∞∼δq,kFprovided thatGholds;
2CqJ,∞∼δq,k−1Fprovided thatG−holds.
Proof. We only prove the case1. Define a bilinear function
au, v λk T
n1
un, vn, ∀u, v∈E. 3.15
Then by2.7we have
|au, v| ≤ λk
λ1uv.
3.16
And hence there exists a unique continuous bounded linear operatorK:E → Esuch that
Ku, vλk T
n1
un, vn. 3.17
SinceKu, u ∈Rfor allu∈E, we can conclude thatKis a self-adjoint operator and
Ju 1
2I−Ku, u −
T
n1
Gun. 3.18
ThenJhas the form2.16with
Qu −
T
n1
Gun, 3.19
and1.11implies that2.17holds. LetA I−K. Then kerA W0 span{φ
k}. Next we
If not, then for anym∈Nand eachαm1/m, there existsumvmwm∈W0⊕W⊕
W−withvm∈W0,wm∈W⊕W−such that
um ≥m, wm ≤
1
mum, 3.20
Jum, vm>0. 3.21
On the other hand,3.20implies
um −→ ∞, vm
um −→
1 as m−→ ∞. 3.22
Thus, byGthere existδ >0 andM∈Nsuch that
T
n1
Gumn, vmn
≥δ, ∀m≥M. 3.23
Therefore,
Jum, vm− T
n1
Gumn, vmn
≤ −δ, ∀m≥M, 3.24
which is a contradiction to 3.21. Consequently AC∞− holds and by Lemma 3.1 and
Proposition 2.5,CqJ,∞ δq,kF. Similarly, we can prove that2holds.
In order to obtain a mountain pass point, we need the following lemmas.
Lemma 3.3. Let
Vt ⎧ ⎨ ⎩
Vt, t≥0,
0, t≤0, V
−t
⎧ ⎨ ⎩
Vt, t≤0,
0, t≥0, 3.25
andV±t t0V±sds. If
lim
|t| → ∞
Vt
t α /λ1, 3.26
then the functional
J±u 1 2
T
n0
|Δun|2−
T
n1
V±un 3.27
Proof. We only prove the caseJ. Let{um} ⊂Esuch that
Jum−→c∈R, Jum−→0 3.28
asm → ∞. SinceEis a finite-dimensional space, it suffices to show that{um}is bounded inE.
Suppose that{um}is unbounded. Passing to a subsequence we may assume thatum → ∞
and for eachn, either|umn| → ∞or{umn}is bounded.
Noticing that for allϕ∈E,
Jum, ϕum, ϕ − T
n1
Vumn, ϕn
. 3.29
Denotewm:um/um, for a subsequence,wmconverges to somewwithw1. By
3.29, we have
Jum, ϕ
um wm, ϕ − T
n1
Vumn
um , ϕn
. 3.30
If|umn| → ∞, then
lim
m→ ∞
Vumn
umn wmn αw
n, 3.31
wherewn max{wn,0}withn∈Z1, T. If{umn}is bounded, then
lim
m→ ∞
Vumn
um 0, wn 0.
3.32
Sincew /0, there is annfor which|umn| → ∞. So passing to the limit in3.30, we have
T
n0
Δwn,Δϕn−α
T
n1
wn, ϕn0. 3.33
This implies thatw /0 satisfies
Δ2wn−1 αwn 0, n∈Z1, T,
w0 0wT1. 3.34
Now, we claim that
In fact, let
wn0 min{wn:n∈Z1, T}. 3.36
We only need to provewn0 > 0. If not, assume thatwn0 ≤ 0. Then by3.34, we have
Δ2wn
0−1 0 and hencewn0−1 wn0 wn01. By induction, it is easy to get
wn 0 for alln∈Z1, Twhich is a contradiction tow /0 and hence3.35holds.
On the other hand, by Proposition 1.1 and Remark 1.2, we see that only the eigenfunction corresponding to the eigenvalue λ1 is positive, which is a contradiction to
α /λ1. The proof is complete.
Lemma 3.4. Under the conditions ofTheorem 1.3, the functionalJ has a critical pointu >0 and CqJ, u∼δq,1F; the functionalJ−has a critical pointu−<0 andCqJ−, u−∼δq,1F.
Proof. We only prove the case of J. Firstly, we prove that J satisfies the Mountain Pass Lemma and henceJhas a nonzero critical pointu. In fact,J∈C1E,Rand byLemma 3.3
we see thatJsatisfies thePScondition. ClearlyJ0 0. Thus we need to show thatJ satisfiesJ1andJ2. To verifyJ1, by1.8andV2, there existρ1 > 0 andρ2 > 0 with
V0< ρ2< λ1such that
Vt≤ 1 2ρ2t
2 3.37
for|t| ≤ρ1. So, for allu∈E, ifu ≤
λ1ρ1, then for eachn∈Z1, T,|un| ≤ρ1and
Ju 1 2u
2−T
n1
Vun
1 2u
2−
n∈N1
Vun
≥ 1 2u
2−1
2ρ2
n∈N1
un, un
≥ 1 2u
2−1
2ρ2
T
n1
un, un
≥ 1 2u
2−1
2 ρ2
λ1u 2
,
3.38
whereN1{n∈Z1, T|un≥0}. Let
ρλ1ρ1, a
1 2
1−ρ2
λ1
ThenJu|∂Bρ≥a >0 and henceJ1holds. ForJ2, byV∞ λk∈λk−1, λk1, we claim that there existγ > λk−1 ≥λ1,b∈Rsuch that
Vt≥ γ 2t
2b, ∀t∈R. 3.40
In fact, by assumption1.9, there existM >0 andb1∈Rsuch thatVt≥γ/2t2b1
for|t| ≥M. Meanwhile, there existsb2 ∈Rsuch thatVt−γ/2t2 ≥b2for|t| ≤Mby virtue
of the continuity ofV. Letbmin{b1, b2}, we get the conclusion.
Thus, if we choosee∈span{φ1}withe >0 ande1, then
Jte t
2
2 −
T
n1
Vten
≤ t2 2 −
γt2
2 e, e−bT
t2 2 −
γt2
2λ1 −bT −→ −∞
3.41
as 0< t → ∞. Thus, we can choose a constanttlarge enough witht > ρandu0 te ∈ E
such thatJu0≤0.J2holds.
Therefore, byTheorem 2.7,J has a critical pointu/0 and similar to the proof of
Lemma 3.3, we can prove thatu >0. Sou is also a critical point ofJ. In the following we compute the critical groupCqJ, uby usingTheorem 2.9.
Assume that
Juv, vv, v −
T
n1
Vunvn, vn≥0, ∀v∈E, 3.42
and that there existsv0/≡0 such that
Juv0, v0, ∀v∈E. 3.43
This implies thatv0satisfies
Δ2v
0n−1 Vunv0n 0, n∈Z1, T,
v00 v0T1 0.
3.44
Hence the eigenvalue problem
Δ2vn−1 λVunvn 0, n∈Z1, T,
v0 vT1 0
has an eigenvalue λ 1. V1 implies that 1 must be a simple eigenvalue; see 4 . So, dim kerJu0 1. Since Eis a finite-dimensional Hilbert space, the Morse index of u
must be finite andJumust be a Fredholm operator. ByTheorem 2.9,CqJ, u∼ δq,1F.
The proof is complete.
Remark 3.5. We can choose the neighborhoodUofusuch thatu >0 for allu∈U. Therefore,
CqJ, u∼CqJ, u∼δq,1F. 3.46
Similarly,
Cq
J, u−∼Cq
J−, u−∼δq,1F. 3.47
Now, we give the proof ofTheorem 1.3.
Proof ofTheorem 1.3. We only prove the casei. ByLemma 3.2,
CqJ,∞∼δq,kF. 3.48
Hence byProposition 2.4the functionalJhas a critical pointu1satisfying
CkJ, u10. 3.49
Since
J0u, u ≥
1−V
0
λ1
u2, 3.50
byV2andJ0 J0 0, we see that 0 is a local minimum ofJ. Hence
CqJ,0∼δq,0F. 3.51
ByRemark 3.5,3.49,3.51, andk ≥ 2 we get thatu,u−,andu1are three nonzero critical
points ofJwithu>0 andu−<0. The proof is complete.
4. An Example and Future Directions
To illustrate the use ofTheorem 1.3, we offer the following example.
Example 4.1. Consider the BVP
Δ2un−1 Vun 0, n∈Z1,5,
u0 0u6,
whereV ∈C2R,Ris defined as follows:
Vt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 10t
2, |t| ≤1,
1 2t
23
4t
4/3, |t| ≥10,
a strictly convex function, otherwise.
4.2
It is easy to verify thatV satisfies1.8,1.9,1.11,V1, andV2withk2. To verify the conditionG, note thatGt t1/3for|t| ≥10, we claim that
5
n1
Gumn, vmn
−→∞, as m−→ ∞ 4.3
which implies thatGholds.
To this end, for any constantr >1, we introduce another norm inET 5as follows:
ur
5
n1
|un|r 1/r
, ∀u∈E. 4.4
SinceEis finite dimensional, there exist two constantsC2≥C1>0 such that
C1u ≤ ur ≤C2u, ∀u∈E. 4.5
Now, byG, for anysmall enough, it is easy to see that
wm ≤um 4.6
holds formlarge enough. Set
Ω1{n∈Z1,5:|umn| ≥10}, Ω2Z1,5\Ω1. 4.7
Sinceum → ∞,Ω1/∅, formlarge enough. And formlarge enough, we have
5
n1
Gumn, vmn
n∈Ω1
u1/3
m n, vmn
n∈Ω2
Gumn, umn
−
n∈Ω2
Gumn, wmn
≥
n∈Ω1
u1m/3n, vmn
−c−cum
n∈Ω1
u1m/3n, umn
−
n∈Ω1
u1m/3n, wmn
−c−cum.
Here and below we denote bycvarious positive constants. Since
n∈Ω1
u1m/3n, umn
5
n1
u1m/3n, umn
−
n∈Ω2
u1m/3n, umn
um44//33−c,
n∈Ω1
u1m/3n, wmn
≤5
n1
|umn|1/3|wmn|
≤ 5
n1
|umn|4/3
1/45
n1
|wmn|4/3
3/4
um14//33wm4/3≤cum44//33.
4.9
Hence
5
n1
Gumn, vmn
≥1−cum44//33−cum −c. 4.10
Since is small enough, we get4.3holds by the above and4.5. Hence, byTheorem 1.3, BVP4.1has at least three nontrivial solutions.
Morse theory has been proved very useful in proving the existence and multiplicity of solutions of operator equations with variational frameworks. However, it is well known that the minimax methods is also a useful tool for the same purpose. The advantage of the minimax methods is that it provides an estimate of the critical value. But it is hard to distinguish critical points obtained by this methods with those by other methods, if the local behavior of the critical points is not very well known. However, critical groups serve as a topological tool in distinguishing isolated critical points. Hence, in order to obtain multiple solutions by using Morse theory, it is crucial to describe critical groups clearly.
A natural question is: can we use the same methods in this paper to other BVPs? Noticing that the key conditions which guarantee the multiplicity of solutions of the BVP 1.1are as follows:
1the BVP has a variational framework;
2the eigenvalues of the corresponding linear BVP are nonzero and there is a one-sign eigenfunction,
hence, if the difference equation
Δ2un−1 Vun 0, n∈Z1, T 4.11
Example 4.2. Consider the BVP
Δ2un−1 Vun 0, n∈Z1, T,
u0 0 ΔuT. 4.12
Let
Eu:u{un}Tn10 with u0 0 ΔuT
. 4.13
ThenEis aT-dimensional Hilbert space with the inner product
u, v
T
n0
Δun,Δvn. 4.14
Define the functionalJonEby
Ju
T
n0
1
2|Δun|
2−T
n1
Vun. 4.15
It is easy to see thatuis a critical point ofJinEif and only ifuis a solution of the BVP4.12. The eigenvalues of the linear BVP
Δ2un−1 λun 0, n∈Z1, T,
u0 0 ΔuT 4.16
are
λλl4 sin2 lπ
22T1, l1,2, . . . , T, 4.17
and the corresponding eigenfunctions are
φln sin lπn
2T1, l1,2, . . . , T. 4.18
Hence,λl/0 for alll ∈ Z1, Tand φ1n > 0 for all n ∈ Z1, T. Therefore, the BVP4.12
satisfies1and2and hence we can obtain similar results as inTheorem 1.3.
However, consider the following difference BVP:
Δ2un−1 Vun 0, n∈Z1, T,
u0 uT, Δu0 ΔuT.
It is easy to verify that the variational functional of the BVP4.19is
Ju
T
n1
1
2|Δun|
2−
Vun
, ∀u∈E1, 4.20
where
E1
u:u{un}Tn01 withu0 uT,Δu0 ΔuT1
. 4.21
But,λ0 is an eigenvalue of the linear BVP:
Δ2un−1 λun 0, n∈Z1, T,
u0 uT, Δu0 ΔuT.
4.22
So, for the BVP4.19, we need to find other techniquese.g., dual variational methods if possibleto study the BVP4.19.
Acknowledgments
The authors would like to express their thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project2009.
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