Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems
© Bo Zheng et al. 2011
Received: 11 November 2010
Accepted: 15 February 2011
Published: 10 March 2011
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.
Let , , and denote the sets of all natural numbers, integers, and real numbers, respectively. For , define , when . is the forward difference operator defined by .
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  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 . 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.  employed the mountain pass lemma to study (1.1) with and obtained the existence of multiple solutions. Very recently, Zheng and Zhang  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.
Hence, grows asymptotically linear at infinity.
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  for details).
and the corresponding eigenfunctions with are .
- (i)The set of functions is orthogonal on with respect to the weight function ; that is,(1.6)
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.
is even and (1.1) has two sign-changing solutions,
is even and (1.1) has six solutions, three of which are of the same sign,
is odd and (1.1) has two sigh-changing solutions,
is odd and (1.1) has three solutions of the same sign.
The assumption in Theorem 1.3 is sharp in the sense that when for , Theorem 1.4 of  gives sufficient conditions for (1.1) to have exactly three solutions with some restrictive conditions.
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.
Here, denotes the Euclidean norm in , and denotes the usual inner product in .
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.
for . The following is the definition of the Palais-Smale (PS) compactness condition.
The functional satisfies the (PS) condition if any sequence such that is bounded and as has a convergent subsequence.
In , Cerami introduced a weak version of the (PS) condition as follows.
The functional satisfies the Cerami (C) condition if any sequence such that is bounded and , as has a convergent subsequence.
for all ,
for all ,
is non-increasing in for any ,
satisfies the (D) condition if satisfies the ( ) condition for all .
Due to the condition , these groups are not dependent on the choice of .
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 ).
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 ).
Moreover, if and , then .
Let denote the open ball in about 0 of the radius , and let denote its boundary.
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 .
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 ).
The following theorem gives a relation between the Leray-Schauder degree and the critical groups.
where denotes the Leray-Schauder degree.
Finally, we state a global version of the Lyapunov-Schmidt reduction method.
Lemma 2.10 (see ).
then the following results hold.
Moreover, is the unique member of such that
(ii) The function : defined by is of class , and
(iii) An element is a critical point of if and only if is a critical point of .
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.
From the definitions of and , it is easy to see that if is a critical point of (or ), then (or ).
The functionals satisfy the (PS) condition; that is, every sequence in such that is bounded, and as has a convergent subsequence.
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.
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 .
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 .
then . And hence, (J1) holds.
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.
The proof is complete.
which implies that 0 is a local minimizer of both and . Hence, (3.21) holds.
Hence, if we set , then (2.11) holds.
Then, (3.23) is proved by Propositions 2.4 and 2.5.
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.
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.
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 .
Now, let . We claim that for large enough and for all , the function has no zero on .
This completes the proof.
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
Now, we can give the proof of Theorem 1.3.
Proof of Theorem 1.3.
Suppose that Is Even
- (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, 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.
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.
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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