Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 172818, 19 pages
doi:10.1155/2011/172818
Research Article
Existence of Positive, Negative, and Sign-Changing
Solutions to Discrete Boundary Value Problems
Bo Zheng,
1
Huafeng Xiao,
1
and Haiping Shi
2
1
School of Mathematics and Information Sciences, Guangzhou University , Guangzhou,
Guangdong 510006, China
2
Department of Basic Courses, Guangdong Baiyun Institute, Guangzhou, Guangdong 510450, China
Correspondence should be addressed to Bo Zheng, [email protected]
Received 11 November 2010; Accepted 15 February 2011
Academic Editor: Zhitao Zhang
Copyright q 2011 Bo Zheng et al. 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.
By using critical point theory, Lyapunov-Schmidt reduction method, and characterization of the
Brouwer degree of critical points, sufficient conditions to guarantee the existence of five or six
solutions together with their sign properties to discrete second-order two-point boundary value
problem are obtained. An example is also given to demonstrate our main result.
1. Introduction
Let , ,and denote the sets of all natural numbers, integers, and real numbers,
respectively. For a, b ∈
T 1
,
1.1
where V ∈ C
2
, ,T≥ 1 is a given integer.
By a solution u to the BVP 1.1, we mean a real sequence {un}
T1
n0
u0,
u1, ,uT 1 satisfying 1.1.Foru {un}
T1
n0
with u00 uT 1,wesay
that u
/
0 if there exists at least one n ∈
1,T such that un
/
0. We say that u is positive
and write u>0 if for all n ∈
1,T,un ≥ 0, and {n ∈ 1,T: un > 0}
/
∅, and similarly,
2 Boundary Value Problems
u is negative u<0 if for all n ∈
1,T,un ≤ 0, and {n ∈ 1,T: un < 0}
/
∅.Wesay
our knowledge, a new method to deal with the sign of solutions in the discrete case.
Here, we assume that V
00and
V
∞
lim
|
t
|
→∞
V
t
t
∈
. 1.2
Hence, V
grows asymptotically linear at infinity.
The solvability of 1.1 depends on the properties of V
both at zero and at infinity. If
V
Δ
2
u
n − 1
λu
n
0,n∈
1,T
,
u
0
0 u
T 1
,
1.4
Boundary Value Problems 3
then we say that 1.1 is resonant at infinity or at zero; otherwise, we say that 1.1 is
nonresonant at infinity or at zero. On the eigenvalue problem 1.4, the following results
hold see 1 for details.
Proposition 1.1. For the eigenvalue problem 1.4, the eigenvalues are
n
0, ∀l
/
j.
1.6
Moreover, for each l ∈
1,T,
T
n1
sin
2
lπn/T 1 T 1/2.
ii It is easy to see that φ
l
is positive and φ
l
changes sign for each l ∈ 2,T;thatis,
{n : φ
l
n > 0}
/
∅ and {n : φ
l
n < 0}
/
∅ for l ∈ 2,T.
0 <λ
k
,λ
k
<V
∞ <λ
k1
for k ∈ 2,T− 1,Theorem1.4of5 gives sufficient
conditions for 1.1 to have exactly three solutions with some restrictive conditions.
Example 1.5. Consider the BVP
Δ
2
u
n − 1
V
u
n
0,n∈
1, 5
,
u
⎪
⎪
⎪
⎪
⎪
⎩
arctan t −
4t
5
,
|
t
|
≤
1
3
,
a strictly increasing function satisfying
1
10
≤ V
t
≤
49
20
,
1
3
,λ
4
,
and 0 <V
t ≤ 49/20 < 3 λ
4
. So, all the conditions in Theorem 1.3 are satisfied with k 3.
And hence 1.7 has at least five solutions, among which two sign-changing solutions or three
solutions of the same sign.
By the computation of critical groups, for k 1, we have the following.
Corollary 1.6 see Remark 3.7 below. If V
0 <λ
1
,V
∞ ∈ λ
1
,λ
2
,and0 <V
t ≤ γ<λ
2
,
then 1.1 has at least one positive solution and one negative solution.
2. Preliminaries
Let
n
,u,v∈ E,
2.2
by which the norm ·can be induced by
u
T
n0
|
Δu
n
|
2
1/2
,u∈ E.
2.3
Here, |·|denotes the Euclidean norm in
,and·, · denotes the usual inner product in .
Define
J
u
J
u
,v
T
n0
Δu
n
, Δv
n
−
T
n1
V
u
,u,v∈ E.
2.5
So, solutions to 1.1 are precisely the critical points of J in E.
As we have mentioned, we will use critical point theory, Lyapunov-Schmidt reduction
method, and degree theory to prove our result. Let us collect some results that will be used
below. One can refer to 10–12 for more details.
Let E be a Hilbert space and J ∈ C
1
E, .Denote
J
c
{
u ∈ E : J
u
≤ c
}
, K
u ∈ E : J
u
0
, K
c
u
m
→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 critical point theory cf. 14, 15.
Definition 2 .3. The functional J satisfies the D
c
condition at the level c ∈ if for any >0
and any neighborhood Nof K
c
,thereare>0 and a continuous deformation η : 0, 1×E →
E such that
i η0,uu for all u ∈ E,
ii ηt, uu for all u/∈ J
−1
c − , c ,
iii Jηt, u is non-increasing in t for any u ∈ E,
iv η1,J
c
\N ⊂ J
c−
.
J satisfies the D condition if J satisfies the D
c
condition for all c ∈ .
Let H
∗
denote singular homology with coefficients in a field .Ifu ∈ E is a critical
J, ∞
H
q
E, J
a
,q∈
0
. 2.8
Due to the condition D
c
, these groups are not dependent on the choice of a.
Assume that #K < ∞ and J satisfies the D condition. The Morse-type numbers of
the pair E, J
a
are defined by M
q
M
q
E, J
a
u∈K
dim C
0
,
2.9
∞
q0
−1
q
M
q
∞
q0
−1
q
β
q
.
2.10
It follows that M
q
≥ β
q
m>0 such th at for all v ∈ X and w
1
,w
2
∈ Y , there holds
∇J
v w
1
−∇J
v w
2
,w
1
− w
2
≥ m
w
1
− w
2
2
,
δ
q,k
.
Let B
r
denote the open ball in E about 0 of the radius r,andlet∂B
r
denote its boundary.
Lemma 2.6 Mountain Pass Lemma 10, 11. Let E be a real Banach space and J ∈ C
1
E,
satisfying the (PS) condition. Suppose that J00 and
Boundary Value Problems 7
J1 there are constants ρ>0,a>0 such that J|
∂B
ρ
≥ a>0,and
J2 thereisau
0
∈ E \ B
ρ
such that Ju
0
≤ 0.
Then, J possesses a critical value c ≥ a.Moreover,c can be characterized as
c inf
h∈Γ
sup
s∈0,1
J
1
J, u 0.
To compute the critical groups of a mountain pass point, we have the following result.
Proposition 2.8 see 11. Let E be a real Hilbert space. Suppose that J ∈ C
2
E, has a mountain
pass point u and that J
u is a Fredholm operator with finite Morse index satisfying
J
u
≥ 0, 0 ∈ σ
J
u
⇒ dim ker
J
u
∞
q0
−1
q
dim C
q
J, u
,
2.17
where d denotes the Leray-Schauder degree.
Finally, we state a global version of the Lyapunov-Schmidt reduction method.
Lemma 2.10 see 18. Let E be a real separable Hilbert space. Let X and Y be closed subspaces of E
such that E X ⊕ Y and J ∈ C
1
E, .Iftherearem>0,α>1 such that for all x ∈ X, y, y
1
∈ Y ,
J
x y
J
x y
.
2.19
Moreover, ψx is the unique member of Y such that
J
x ψ
x
,y
0, ∀y ∈ Y. 2.20
ii The function
J : X → defined by JxJx ψx is of class C
1
,and
J
x
,x
x ψ
x
: x ∈ X
. 2.22
If
J
x
/
0 for x ∈ ∂S,then
d
J
,S,0
d
J
, Σ, 0
, 2.23
where d denotes the Leray-Schauder degree.
v If u x ψx is a critical p oint of mountain pass type of J,thenx is a critical point of
mountain pass type of
Based on these results, four nontrivial solutions {u
,u
−
,u
0
,u
1
} to 1.1 can be obtained by
2.9 or 2.10. However, it seems difficult to obtain the sign property of u
0
and u
1
through
their depiction of critical groups. To conquer this difficulty, we compute the Brouwer degree
of the s ets of positive solutions and negative solutions to 1.1. Finally, the third nontrivial
solution to 1.1 is obtained by Lyapunov-Schmidt reduction method, and its characterization
of the local degree results in one or two more nontrivial solutions to 1.1 together with their
sign property.
Boundary Value Problems 9
Let
V
t
⎧
⎨
⎩
0
t, t > 0,
3.1
and V
±
x
x
0
V
±
sds. The functionals J
±
: E → are defined as
J
±
u
1
2
u
2
−
T
m
is bounded, and J
±
u
m
→ 0 as m →∞has a convergent subsequence.
Proof. We only prove the case of J
.ThecaseofJ
−
is completely similar. Since E is finite
dimensional, it suffices to show that {u
m
} is bounded. Suppose that {u
m
} is unbounded.
Passing to a subsequence, we may assume that u
m
→∞and for each n,either|u
m
n|→
∞ or {u
m
n} is bounded.
Set w
m
u
m
/u
m
n1
V
u
m
n
,ϕ
n
.
3.3
Hence,
J
u
m
,ϕ
u
u
m
,ϕ
n
.
3.4
If |u
m
n|→∞,then
lim
m →∞
V
u
m
n
u
m
lim
m →∞
0
w
−
n
,
3.5
where w
nmax{wn, 0}, w
−
nmin{wn, 0}.If{u
m
n} is bounded, then
lim
m →∞
V
u
m
n
u
m
∞
w
n
V
0
w
−
n
,ϕ
n
0,
3.7
which implies that wn satisfies
Δ
2
w
n −1
0 w
T 1
.
3.8
Because V
0 <λ
1
,weseethatifw
/
0 is a solution to 3.8,thenu is positive. Since this
contradicts V
∞ ∈ λ
k
,λ
k1
,weconcludethatw ≡ 0 is the only solution to 3.8.A
contradiction to w 1.
Lemma 3.3. Under the conditions of Theorem 1 .3, J
has a positive mountain pass-type critical point
u
with C
q
J
J, u
−
∼
δ
q,1
.
Proof. We only prove the case of J
. Firstly, we will prove that J
satisfies all the conditions
in Lemma 2.6. And hence, J
has at least one nonzero critical point u
.Infact,J
∈ C
1
E, ,
and J
satisfies the PS condition by Lemma 3.2. Clearly, J
00. Thus, we still have to
show that J
satisfies J1, J2.ToverifyJ1,setα : V
1
2
α
t
2
, for
|
t
|
≤ ρ
1
.
3.10
Take λ
1
− α/2 > 0, then α λ
1
α/2 ∈ α, λ
1
.Ifwesetρ
2
λ
1
α/2, then
V
t
1
2
u
2
−
T
n1
V
u
n
1
2
u
2
−
n∈N
1
V
2
ρ
2
n∈N
1
u
n
,u
n
−
1
2
α
n∈N
2
u
n
,u
n
2
−
1
2
ρ
2
λ
1
u
2
,
3.12
where N
1
{n ∈ 1,T | un ≥ 0}, N
2
{n ∈ 1,T | un < 0}.Ifwetake
ρ
λ
1
ρ
1
,a
1
2
t
≥
γ
2
t
2
b, for t ∈ .
3.14
So, if we take φ
1
n > 0withφ
1
1, then
J
tφ
1
t
2
2
−
T
n1
V
− bT −→ −∞, 0 <t−→ ∞.
3.15
So, if we take t sufficiently large such that t>ρand for u
0
te ∈ E, J
u
0
≤ 0, then J2
holds.
Now, by Lemma 2.6, J
has at least a nonzero critical point u
.Andforalln ∈ 1,T,
we claim that u
n ≥ 0. If not, set A
1
{n ∈ 1,T | u
n < 0},thenforalln ∈ A
1
, Δ
2
u
n−
1V
v, v
v, v
−
T
n1
V
u
n
v
n
,v
n
≥ 0, ∀v ∈ E,
3.16
n
v
0
n
0,n∈
1,T
,
v
0
0
v
0
T 1
0.
3.18
Hence, the eigenvalue problem
Δ
2
v
has an eigenvalue λ 1. Condition V
t > 0 implies that 1 must be a simple eigenvalue;
see 1.So,dimkerJ
u
1. Since E is finite dimensional, the Morse index of u
must be
finite and J
u
must be a Fredholm operator. By Proposition 2.8, C
q
J, u
∼
δ
q,1
. Finally,
choose the neighborhood U
of u
such that u>0forallu ∈ U
J, 0
∼
C
q
J
, 0
∼
δ
q,0
. 3.21
Proof. By assumption, we have J
0J
00andforallu ∈ E \{0},
J
0
u, u
1 −
V
0
λ
1
u
2
> 0,
3.22
which implies that 0 is a local minimizer of both J
and J.Hence,3.21 holds.
Remark 3.5. Under the conditions of Theorem 1.3,wehave
C
q
J, ∞
∼
δ
q,k
. 3.23
Boundary Value Problems 13
x y
− J
x y
1
,y− y
1
y − y
1
2
−
T
n1
V
ξ
n
2
≥
y − y
1
2
−
γ
λ
k1
y − y
1
2
.
3.25
Hence, if we set m 1 − γ/λ
k1
,then2.11 holds.
Now, noticing that V
Hence, we have
J
u
−→ −∞, as u ∈ X,
u
−→ ∞, 3.27
J
u
−→ ∞, as u ∈ Y,
u
−→ ∞. 3.28
Then, 3.23 is proved by Propositions 2.4 and 2.5.
Remark 3.6. Following the proof of Theorem 3.1 in 17, 3.23 implies that there must exist a
critical point u
0
/
0ofJ satisfying
C
q
J, u
0
1
× 2
−1
k
−1
k
.
3.30
This is impossible. Thus, J must have at least one more critical point u
1
.Hence,1.1 has
at least five solutions. However, it seems difficult to obtain the sign property of u
0
and u
1
.
14 Boundary Value Problems
To obtain more refined results, we seek the third nontrivial solution u
0
to 1.1 by Lyapunov-
Schmidt reduction method and then its characterization of the local degree results in one or
two more nontrivial solutions to 1.1 together with their sign property.
Remark 3.7. The condition k ≥ 2inTheorem 1.3 is necessary to obtain three or more nontrivial
solutions to 1.1.Infact,ifk 1, then we have
C
or u
−
which becomes an obstacle to seek other critical points
by using Morse inequality. If k 0, then
C
q
J, u
0
∼
C
q
J, 0
∼
δ
q,0
. 3.32
Hence, one cannot exclude the possibility of u
0
0.
To compute the degree of the set of positive or negative solutions to 1.1, we need
the following lemma.
Lemma 3.8. There exists ρ>0 large enough, such that
d
⎧
⎨
⎩
γ
1
t, t ≥ 0,
V
0
t, t < 0.
3.34
Let Pt
t
0
psds. The functional Q : E → is defined as
Q
u
1
2
u
2
−
,
u
0
0 u
T 1
.
3.36
Since V
0 <λ
1
,weseethatifu
/
0 is a solution to 3.36,thenu is positive. Because this
contradicts γ>λ
1
,weconcludethatu ≡ 0 is the only critical point of Q.
Boundary Value Problems 15
We claim that if B is a ball in E containing zero, then dQ
,B,00. In fact, since
γ
1
>λ
1
>V
−
T
n1
λ
1
u
n
,φ
1
n
−
T
n1
h
u
n
,φ
1
n1
h
u
n
,φ
1
n
−
T
n1
h
u
n
,φ
1
n
,B,0
d
K, B, 0
0, 3.39
where Ku−φ
1
.
Now, let γ
1
V
∞. We claim that for ρ>0 large enough and for all s ∈ 0, 1,the
function sJ
1 − sQ
has no zero on ∂B
ρ
.
In fact, we have proved that for all ρ>0andforallu ∈ ∂B
ρ
,wehave
Q
t
t
>
V
∞
λ
k
2
>λ
k
.
3.41
For t ≤−ρ,take
2
λ
k1
− V
∞/2, then
V
t
q
t
: V
t
− λ
1
t
⎧
⎨
⎩
V
t
− λ
1
t, t ≥ 0,
V
0
t − λ
1
−
T
n1
V
u
n
,φ
1
n
−
T
n1
Δ
2
φ
1
n − 1
φ
1
n
,u
n
−
T
n1
V
u
n
,φ
1
n
−
T
, 0
d
Q
,B
ρ
, 0
0. 3.45
This completes the proof.
Remark 3.9. By Theorem 2.9 and the above results, we have the following characterization of
degree of critical points.
i If U
U
−
is a neighborhood of u
u
−
containing no other critical points, then
d
J
,U
, 0
J
,B,0
1 d
J
,B,0
. 3.47
Hence, if Σ is a bounded region containing the positive critical points and no other
critical points, then by 3.33 we have
d
J
, Σ, 0
d
J
, Σ, 0
d
J
, 0
−1. 3.49
Boundary Value Problems 17
Now, we can give the proof of Theorem 1.3.
Proof of Theorem 1.3. The functional J satisfies 2.18 in Lemma 2.10 due to the fact that J
satisfies 2.11.Hence,byLemma 2.10,thereexistsψ : X →
such that
J
x ψ
x
min
y∈Y
J
x y
.
3.50
Moreover, ψx is the unique member of Y such that
J
x ψ
0
containing no other critical points of J, taking W {x ∈ X : x ψx ∈
V },thend
J
,W,0−1
k
. Then, by part iv of Lemma 2.10,wehave
d
J
,V,0
−1
k
.
3.52
Suppose that k Is Even
Let R
1
be large enough so that if J
x0, then x <R
1
.BecausedimX<∞ and ψx is of
class C
J
,B
R
1
, 0
−1
k
1. 3.53
Suppose that K {u ∈ E | J
u0} is finite. Let S
1
, S
2
,andS
3
be disjoint open bounded
regions in E such that
S
1
∩K {0}, S
2
∩Kis the set of positive critical points of J,andS
3
∩K
ψx
0
/∈ S
2
∪S
3
,thenu
0
is sign changing. Let S
4
denote an open bounded
region disjoint from
S
1
∪ S
2
∪ S
3
such that S
4
∩K {u
0
}. By the excision property of Brouwer
degree, we have
−1 d
J
,C,0
, 0
d
J
,C− S
1
∪ S
2
∪ S
3
∪ S
4
, 0
1 − 1 − 1 1 d
J
,C− S
1
∪ S
2
∪ S
3
∪ S
4
, 0
3
. Without loss of generality, we may assume that
u
0
∈ S
2
.LetS
4,2
be a neighborhood of u
0
such that S
4,2
∩K {u
0
}.ByLemma 3.3,thereexists
a critical point of mountain pass type u
∈ S
2
such that if S
5
is a neighborhood of u
such that
S
5
∩K {u
},thendJ
,S
2
− S
4,2
∪ S
5
, 0
1 − 1 d
J
,S
2
− S
4,2
∪ S
5
, 0
.
3.56
Thus, by Kronecker existence property of Brouwer degree, there exists u
1
∈ S
2
−S
4,2
∪ S
J
,S
3
, 0
d
J
,C− S
1
∪ S
2
∪ S
3
, 0
1 − 1 − 1 d
J
,C− S
1
∪ S
2
∪ S
3
, 0
2
/∈ S
2
∪ S
3
and u
,u
0
,u
1
∈ S
2
, u
2
is a sign-changing solution, and u
, u
0
,andu
1
have the
same sign. This completes the proof of Theorem 1.3,whenk is even.
Suppose that k Is Odd
iii Let S
1
, S
2
,andS
3
and x
0
is
a local maximum of
J
. Since we are assuming 1.1 to have only finitely many solutions, x
0
is
a strictly local maximum of
J
.Letδ>0besuchthatJ
x < J
x
0
if 0 < x − x
0
<δ.Since
k ≥ 2, {x :0< x − x
0
<} is path connected. Thus, x
0
is not a critical point of mountain
pass type. By Lemma 3.3, J
has a critical point of mountain pass type u
} and V
1
∩K {u
}.Thus,
−1 d
J
,S
2
, 0
d
J
,V
0
, 0
d
J
,V
1
, 0
d
2
− V
0
∪ V
1
. So far, we have proved that 1.1 has at least four nontrivial solutions
{u
−
,u
,u
0
,u
1
} and that u
,u
0
,u
1
∈ S
2
have the same sign. This proves Theorem 1.3.
Acknowledgments
Project supported by National Natural Science Foundation of China no. 11026059 and
Foundation for Distinguished Young Talents in Higher Education of Guangdong, China no.
LYM09105.
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