- Research Article
- Open access
- Published:
Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems
Boundary Value Problems volume 2011, Article number: 172818 (2011)
Abstract
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
, define
, when
.
is the forward difference operator defined by
.
Consider the following discrete second-order two-point boundary value problem (BVP for short):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ1_HTML.gif)
where is a given integer.
By a solution to the BVP (1.1), we mean a real sequence
satisfying (1.1). For
with
, we say that
if there exists at least one
such that
. We say that
is positive (and write
) if for all
, and
 : 
, and similarly,
is negative (
) if for all
, and
. We say that
is sign-changing if
is neither positive nor negative. Under convenient assumptions, we will prove the existence of five or six solutions to (1.1), which include positive, negative, and sign-changing solutions.
Difference BVP has widely occurred as the mathematical models describing real-life situations in mathematical physics, finite elasticity, combinatorial analysis, and so forth; for example, see [1, 2]. And many scholars have investigated difference BVP independently mainly for two reasons. The first one is that the behavior of discrete systems is sometimes sharply different from the behavior of the corresponding continuous systems. For example, every solution of logistic equation is monotone, but its discrete analogue
has chaotic solutions; see [3] for details. The second one is that there is a fundamental relationship between solutions to continuous systems and the corresponding discrete systems by employing discrete variable methods [1]. The classical results on difference BVP employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point theorems. We remark that, usually, the application of the fixed point theorems yields existence results only.
Recently, however, a few scholars have used critical point theory to deal with the existence of multiple solutions to difference BVP. For example, in 2004, Agarwal et al. [4] employed the mountain pass lemma to study (1.1) with and obtained the existence of multiple solutions. Very recently, Zheng and Zhang [5] obtained the existence of exactly three solutions to (1.1) by making use of three-critical-point theorem and analytic techniques. We also refer to [6–9] for more results on the difference BVP by using critical point theory. The application of critical point theory to difference BVP represents an important advance as it allows to prove multiplicity results as well.
Here, by using critical point theory again, as well as Lyapunov-Schmidt reduction method and degree theory, a sharp condition to guarantee the existence of five or six solutions together with their sign properties to (1.1) is obtained. And this paper offers, to the best of our knowledge, a new method to deal with the sign of solutions in the discrete case.
Here, we assume that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ2_HTML.gif)
Hence, grows asymptotically linear at infinity.
The solvability of (1.1) depends on the properties of both at zero and at infinity. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ3_HTML.gif)
where is one of the eigenvalues of the eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ5_HTML.gif)
and the corresponding eigenfunctions with are
.
Remark 1.2.
-
(i)
The set of functions
is orthogonal on
with respect to the weight function
; that is,
(1.6)
Moreover, for each .
-
(ii)
It is easy to see that
is positive and
changes sign for each
; that is,
 : 
and
 : 
for
.
The main result of this paper is as follows.
Theorem 1.3.
If with
, and
, then (1.1) has at least five solutions. Moreover, one of the following cases occurs:
-
(i)
  
is even and (1.1) has two sign-changing solutions,
-
(ii)
  
is even and (1.1) has six solutions, three of which are of the same sign,
-
(iii)
  
is odd and (1.1) has two sigh-changing solutions,
-
(iv)
  
is odd and (1.1) has three solutions of the same sign.
Remark 1.4.
The assumption in Theorem 1.3 is sharp in the sense that when
for
, Theorem 1.4 of [5] gives sufficient conditions for (1.1) to have exactly three solutions with some restrictive conditions.
Example 1.5.
Consider the BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ7_HTML.gif)
where is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ8_HTML.gif)
It is easy to verify that ,
, and
. So, all the conditions in Theorem 1.3 are satisfied with
. 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 , we have the following.
Corollary 1.6 (see Remark 3.7 below).
If , and
, then (1.1) has at least one positive solution and one negative solution.
2. Preliminaries
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ9_HTML.gif)
Then, is a
-dimensional Hilbert space with inner product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ10_HTML.gif)
by which the norm can be induced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ11_HTML.gif)
Here, denotes the Euclidean norm in
, and
denotes the usual inner product in
.
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ12_HTML.gif)
Then, the functional is of class
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ13_HTML.gif)
So, solutions to (1.1) are precisely the critical points of in
.
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 be a Hilbert space and
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ14_HTML.gif)
for . The following is the definition of the Palais-Smale (PS) compactness condition.
Definition 2.1.
The functional satisfies the (PS) condition if any sequence
such that
is bounded and
as
has a convergent subsequence.
In [13], Cerami introduced a weak version of the (PS) condition as follows.
Definition 2.2.
The functional satisfies the Cerami (C) condition if any sequence
such that
is bounded and
, as
has a convergent subsequence.
If satisfies the (PS) condition or the (C) condition, then
satisfies the following deformation condition which is essential in critical point theory (cf. [14, 15]).
Definition 2.3.
The functional satisfies the (
) condition at the level
if for any
and any neighborhood of
, there are
and a continuous deformation
such that
-
(i)
  
for all
,
-
(ii)
  
for all
,
-
(iii)
   
is non-increasing in
for any
,
-
(iv)
  
.
satisfies the (D) condition if
satisfies the (
) condition for all
.
Let denote singular homology with coefficients in a field
. If
is a critical point of
with critical level
, then the critical groups of
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ15_HTML.gif)
Suppose that is strictly bounded from below by
and that
satisfies (
) for all
. Then, the
th critical group at infinity of
is defined in [16] as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ16_HTML.gif)
Due to the condition , these groups are not dependent on the choice of
.
Assume that and
satisfies the (D) condition. The Morse-type numbers of the pair
are defined by
, and the Betti numbers of the pair
are defined by
. By Morse theory [10, 11], the following relations hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ18_HTML.gif)
It follows that for all
. If
, then
for all
. Thus, when
for some
must have a critical point
with
.
The critical groups of at an isolated critical point
describe the local behavior of
near
, while the critical groups of
at infinity describe the global property of
. In most applications, unknown critical points will be found from (2.9) or (2.10) if we can compute both the critical groups at known critical points and the critical groups at infinity. Thus, the computation of the critical groups is very important. Now, we collect some useful results on computation of critical groups which will be employed in our discussion.
Proposition 2.4 (see [16]).
Let be a real Hilbert space and
. Suppose that
splits as
such that
is bounded from below on
and
for
as
. Then
for
.
Proposition 2.5 (see [17]).
Let be a separable Hilbert space with inner product
and corresponding norm
,
,
closed subspaces of
such that
. Assume that
satisfies the (PS) condition and the critical values of
are bounded from below. If there is a real number
such that for all
and
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ19_HTML.gif)
then there exists a -functional
 : 
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ20_HTML.gif)
Moreover, if and
, then
.
Let denote the open ball in
about 0 of the radius
, and let
denote its boundary.
Lemma 2.6 (Mountain Pass Lemma [10, 11]).
Let be a real Banach space and
satisfying the (PS) condition. Suppose that
and
(J1)  there are constants such that
, and
(J2)  there is a such that
.
Then, possesses a critical value
. Moreover,
can be characterized as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ21_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ22_HTML.gif)
Definition 2.7 (Mountain pass point).
An isolated critical point of
is called a mountain pass point if
.
To compute the critical groups of a mountain pass point, we have the following result.
Proposition 2.8 (see [11]).
Let be a real Hilbert space. Suppose that
has a mountain pass point
and that
is a Fredholm operator with finite Morse index satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ23_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ24_HTML.gif)
The following theorem gives a relation between the Leray-Schauder degree and the critical groups.
Let be a real Hilbert space, and let
be a function satisfying the (PS) condition. Assume that
, where
 : 
is a completely continuous operator. If
is an isolated critical point of
, that is, there exists a neighborhood
of
, such that
is the only critical point of
in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ25_HTML.gif)
where denotes the Leray-Schauder degree.
Finally, we state a global version of the Lyapunov-Schmidt reduction method.
Lemma 2.10 (see [18]).
Let be a real separable Hilbert space. Let
and
be closed subspaces of
such that
and
. If there are
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ26_HTML.gif)
then the following results hold.
(i)  There exists a continuous function  : 
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ27_HTML.gif)
Moreover, is the unique member of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ28_HTML.gif)
(ii)  The function  : 
defined by
is of class
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ29_HTML.gif)
(iii)  An element is a critical point of
if and only if
is a critical point of
.
(iv)  Let and
be the projection onto
across
. Let
and
be open bounded regions such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ30_HTML.gif)
If for
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ31_HTML.gif)
where denotes the Leray-Schauder degree.
(v)  If is a critical point of mountain pass type of
, then
is a critical point of mountain pass type of
.
3. Proof of Theorem 1.3
In this section, firstly, we obtain a positive solution and a negative solution
with
to (1.1) by using cutoff technique and the mountain pass lemma. Then, we give a precise computation of
. And we remark that under the assumptions of Theorem 1.3,
can be completely computed by using Propositions 2.4 and 2.5. Based on these results, four nontrivial solutions
to (1.1) can be obtained by (2.9) or (2.10). However, it seems difficult to obtain the sign property of
and
through their depiction of critical groups. To conquer this difficulty, we compute the Brouwer degree of the sets 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.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ32_HTML.gif)
and . The functionals
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ33_HTML.gif)
Remark 3.1.
From the definitions of and
, it is easy to see that if
is a critical point of
(or
), then
(or
).
Lemma 3.2.
The functionals satisfy the (PS) condition; that is, every sequence
in
such that
is bounded, and
as
has a convergent subsequence.
Proof.
We only prove the case of . The case of
is completely similar. Since
is finite dimensional, it suffices to show that
is bounded. Suppose that
is unbounded. Passing to a subsequence, we may assume that
and for each
, either
or
is bounded.
Set . For a subsequence,
converges to some
with
. Since for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ34_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ35_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ36_HTML.gif)
where ,
. If
is bounded, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ37_HTML.gif)
Letting in (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ38_HTML.gif)
which implies that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ39_HTML.gif)
Because , we see that if
is a solution to (3.8), then
is positive. Since this contradicts
, we conclude that
is the only solution to (3.8). A contradiction to
.
Lemma 3.3.
Under the conditions of Theorem 1.3, has a positive mountain pass-type critical point
with
has a negative mountain pass-type critical point
with
.
Proof.
We only prove the case of . Firstly, we will prove that
satisfies all the conditions in Lemma 2.6. And hence,
has at least one nonzero critical point
. In fact,
, and
satisfies the (PS) condition by Lemma 3.2. Clearly,
. Thus, we still have to show that
satisfies (J1), (J2). To verify (J1), set
, then for any
, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ40_HTML.gif)
So, by Taylor series expansion,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ41_HTML.gif)
Take , then
. If we set
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ42_HTML.gif)
Since for all , if
, then
for every
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ43_HTML.gif)
where ,
. If we take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ44_HTML.gif)
then . And hence, (J1) holds.
To verify (J2), note that implies that there exist
and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ45_HTML.gif)
So, if we take with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ46_HTML.gif)
So, if we take sufficiently large such that
and for
, then (J2) holds.
Now, by Lemma 2.6, has at least a nonzero critical point
. And for all
, we claim that
. If not, set
, then for all
. By
for all
. Hence,
.
In the following, we will compute the critical groups by using Proposition 2.8.
Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ47_HTML.gif)
and that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ48_HTML.gif)
This implies that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ49_HTML.gif)
Hence, the eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ50_HTML.gif)
has an eigenvalue . Condition
implies that 1 must be a simple eigenvalue; see [1]. So,
. Since
is finite dimensional, the Morse index of
must be finite and
must be a Fredholm operator. By Proposition 2.8,
. Finally, choose the neighborhood
of
such that
for all
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ51_HTML.gif)
The proof is complete.
Lemma 3.4.
By , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ52_HTML.gif)
Proof.
By assumption, we have and for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ53_HTML.gif)
which implies that 0 is a local minimizer of both and
. Hence, (3.21) holds.
Remark 3.5.
Under the conditions of Theorem 1.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ54_HTML.gif)
We will use Propositions 2.4 and 2.5 to prove (3.23). Very similar to the proof of Lemma 3.2, we can prove that satisfies the (PS) condition. And it is easy to prove that
satisfies (2.11). In fact, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ55_HTML.gif)
By , for all
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ56_HTML.gif)
Hence, if we set , then (2.11) holds.
Now, noticing that implies that there exist
,
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ57_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ59_HTML.gif)
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 of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ60_HTML.gif)
It is known that the critical groups are useful in distinguishing critical points. So far, we have obtained four critical points 0, ,
, and
together with their characterization of critical groups. Assume that 0,
,
, and
are the only critical points of
. Then, the Morse inequality (2.10) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ61_HTML.gif)
This is impossible. Thus, must have at least one more critical point
. Hence, (1.1) has at least five solutions. However, it seems difficult to obtain the sign property of
and
. To obtain more refined results, we seek the third nontrivial solution
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 in Theorem 1.3 is necessary to obtain three or more nontrivial solutions to (1.1). In fact, if
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ62_HTML.gif)
Hence, may coincide with
or
which becomes an obstacle to seek other critical points by using Morse inequality. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ63_HTML.gif)
Hence, one cannot exclude the possibility of .
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 large enough, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ64_HTML.gif)
Proof.
We only prove the case of . For any
, define
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ65_HTML.gif)
Let . The functional
 : 
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ66_HTML.gif)
It is obvious that is of class
and its critical points are precisely solutions to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ67_HTML.gif)
Since , we see that if
is a solution to (3.36), then
is positive. Because this contradicts
, we conclude that
is the only critical point of
.
We claim that if is a ball in
containing zero, then
. In fact, since
and
for
. Hence, for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ68_HTML.gif)
where we have used the fact that is positive on
. Then, for each
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ69_HTML.gif)
Hence, by invariance under homotopy of Brouwer degree, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ70_HTML.gif)
where .
Now, let . We claim that for
large enough and for all
, the function
has no zero on
.
In fact, we have proved that for all and for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ71_HTML.gif)
On the other hand, by the definition of , for all
, there exists
large enough such that
for
. Since
, for
, take
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ72_HTML.gif)
For , take
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ73_HTML.gif)
Hence, if we take , then for
, we have
, and for
, we have
. So, if we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ74_HTML.gif)
then for all
. And for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ75_HTML.gif)
So far, we have proved that for large enough,
has no zero point on
for each
. Hence, by invariance under homotopy of Brouwer degree, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ76_HTML.gif)
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.
  If (
) is a neighborhood of
(
) containing no other critical points, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ77_HTML.gif)
  Assume that is a ball centered at zero containing on other critical points, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ78_HTML.gif)
Hence, if is a bounded region containing the positive critical points and no other critical points, then by (3.33) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ79_HTML.gif)
Similarly, we see that if is a bounded region containing the negative critical points and no other critical points, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ80_HTML.gif)
Now, we can give the proof of Theorem 1.3.
Proof of Theorem 1.3.
The functional satisfies (2.18) in Lemma 2.10 due to the fact that
satisfies (2.11). Hence, by Lemma 2.10, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ81_HTML.gif)
Moreover, is the unique member of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ82_HTML.gif)
The function defined by
is of class
. Because
, (3.27) implies that
as
. Since
, there must exist
such that
. Take
, then
by (iii) of Lemma 2.10. If
is a neighborhood of
containing no other critical points of
, taking
, then
. Then, by part (iv) of Lemma 2.10, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ83_HTML.gif)
Suppose that Is Even
Let be large enough so that if
, then
. Because
and
is of class
, there exists
such that
for
. Because
is coercive,
. Hence, if we set
 : 
, then by (iv) of Lemma 2.10, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ84_HTML.gif)
Suppose that is finite. Let
,
, and
be disjoint open bounded regions in
such that
is the set of positive critical points of
, and
is the set of negative critical points of
. So far, we have proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ85_HTML.gif)
(i) If , then
is sign changing. Let
denote an open bounded region disjoint from
such that
. By the excision property of Brouwer degree, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ86_HTML.gif)
Thus, by Kronecker existence property of Brouwer degree, we see that there must exist such that
, which proves that (1.1) has at least five solutions. In this case, both
and
change sign.
-
(ii)
Suppose now that
. Without loss of generality, we may assume that
. Let
be a neighborhood of
such that
. By Lemma 3.3, there exists a critical point of mountain pass type
such that if
is a neighborhood of
such that
, then
. Thus,
(3.56)
Thus, by Kronecker existence property of Brouwer degree, there exists such that
. Finally,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ88_HTML.gif)
Thus, there must exist such that
. Thus, the set
together with a critical point
of
in
shows that (1.1) has five nontrivial solutions. Since
and
,
is a sign-changing solution, and
,
, and
have the same sign. This completes the proof of Theorem 1.3, when
is even.
Suppose that Is Odd
-
(iii)
Let
,
, and
be as above. If
, the proof follows very closely that of the case (i).
-
(iv)
Suppose that
, hence
. Because
, there exists
such that
if
. So, if
, then
and
is a local maximum of
. Since we are assuming (1.1) to have only finitely many solutions,
is a strictly local maximum of
. Let
be such that
if
. Since
,
is path connected. Thus,
is not a critical point of mountain pass type. By Lemma 3.3,
has a critical point of mountain pass type
. By (v) of Lemma 2.10,
, and hence
. Let
be neighborhoods of
and
, respectively, such that
and
. Thus,
(3.58)
Thus, by Kronecker existence property of Brouwer degree, there exists a third positive solution . So far, we have proved that (1.1) has at least four nontrivial solutions
and that
have the same sign. This proves Theorem 1.3.
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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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Zheng, B., Xiao, H. & Shi, H. Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems. Bound Value Probl 2011, 172818 (2011). https://doi.org/10.1155/2011/172818
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DOI: https://doi.org/10.1155/2011/172818