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Multiplicity results on discrete boundary value problems with double resonance via variational methods
Advances in Difference Equations volume 2013, Article number: 309 (2013)
The existence of solutions for a class of difference equations with double resonance is studied via variational methods, and multiplicity results are derived.
Let Z denote the set of integers, and for with , define . For a given positive integer , consider the following discrete boundary value problem:
where Δ is the forward difference operator defined by and for . Throughout this paper, we always assume that is -differentiable with respect to the second variable and satisfies for , which implies that (BP) has a trivial solution , . We investigate the existence of nontrivial solutions of (BP).
In different fields of research, such as computer science, mechanical engineering, control systems, population biology, economics and many others, the mathematical modeling of important questions leads naturally to the consideration of nonlinear difference equations. The dynamic behaviors of nonlinear difference equations have been studied extensively in [1, 2]. Recently, many authors considered the solvability of nonlinear difference equations via variational methods. For example, on the second-order difference equations, the boundary value problems are studied in [3–7] and the existence of periodic solutions is investigated in [8–10].
As a natural phenomenon, resonance exists in the real world from macrocosm to microcosm. In a system described by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity. It is known from  that the eigenvalue problem
possesses N distinct eigenvalues , . Many authors considered the complete resonance situation in the sense that for some ,
via different methods in critical point theory such as Morse theory , index theory  and minimax methods . The assumption that
() there exists some such that
characterizes problem (BP) as double resonance between two consecutive eigenvalues at infinity. In the case of resonance, one needs to impose various conditions on the nonlinearity of f near infinity to ensure the global compactness. In fact, many results on differential equations with double resonance have been obtained (see [11–13]). As to discrete boundary value problems with double resonance, however, there are few results published. In , the existence of periodic solutions to a second-order difference equation with double resonance, as is described in (), is investigated.
Motivated by the study in , we consider problem (BP) with double resonance indicated in (). To control the double resonance, a selectable restriction on the nonlinearity of f is that
() there exists some such that
which has completely the same form as its counterpart in . However, instead of (), in this paper we assume that
(f∞) there exists some such that
Remark 1.1 It is easy to see that, as a restriction on the nonlinearity of f, (f∞) is more relaxed than () (see Examples 1.1-1.3 and Remark 1.3). In addition, (f∞), as well as (), implies ().
A sequence is said to be a positive (negative) solution of (BP) if it satisfies (BP) and (<0) for .
Theorem 1.1 Assume that (f∞) holds. Then (BP) has at least four nontrivial solutions in which one is positive and one is negative in each of the following two cases:
and for ;
and for .
To state the following theorems, we further assume that
(f0) there exists such that for .
Theorem 1.2 Assume that (f0) and (f∞) hold with . If there exists with such that for , then (BP) has at least four nontrivial solutions.
Let denote the derivative of with respect to the second variable. In the case where (BP) is also resonant at the origin, that is, there exists such that for , we assume that
() for small and .
Theorem 1.3 Assume that (f0) and (f∞) hold with . If there exists such that for , then (BP) has at least four nontrivial solutions in each of the following two cases:
() with and ;
() with and .
Remark 1.2 In view of the proofs in Section 4, we see that if (<0) in (f0), two of the solutions derived in Theorems 1.2, 1.3 are positive (negative).
Set and define by
By calculation, we get and . Define , and , , where α and β are constants. Obviously, , and , . The following examples are presented to illustrate the applications of the above results.
Example 1.1 Consider (BP) with , . We have for . If and or and , then by Theorem 1.1, (BP) has at least four nontrivial solutions in which one is positive and one is negative.
Example 1.2 Set with . Let and . Consider (BP) with , . We have and for , which implies that there exists such that for . By Theorem 1.2 and Remark 1.2, (BP) has at least four nontrivial solutions in which two are positive.
Example 1.3 Set , and for some . Consider (BP) with , . We have and for , which implies that there exists such that for . Moreover,
By Theorem 1.3(ii) and Remark 1.2, (BP) has at least four nontrivial solutions in which two are positive.
Remark 1.3 It is easy to see that , , that is, the restriction imposed here is more relaxed than that in .
The paper is organized as follows. In Section 2 we give a simple revisit to Morse theory, and in Section 3 we give some lemmas. The main results will be proved in Section 4.
2 Preliminary results on critical groups
Let H be a Hilbert space and be a functional satisfying the Palais-Smale condition ((PS) in short), that is, every sequence such that is bounded and as has a convergent subsequence. Denote by the q th singular relative homology group of the topological pair with integer coefficients. Let be an isolated critical point of Φ with , , and U be a neighborhood of . For , the group
is called the q th critical group of Φ at , where .
If the set of the critical points of Φ, denoted by , is finite and , the critical groups of Φ at infinity are defined by (see )
For , we call the Betti numbers of Φ and define the Morse-type numbers of the pair by
With the above notations, we have the following facts (2.a)-(2.f) [, Chapter 8].
(2.a) If for some , then there exists such that ;
(2.b) If , then ;
If and is a Fredholm operator and the Morse index and nullity of are finite, then we have
(2.d) for ;
(2.e) If , then , and if , then ;
(2.f) If , then when is local minimizer of Φ, while when is the local maximizer of Φ.
We say that Φ has a local linking at if there exists the direct sum decomposition: and such that
The following results are due to Su .
(2.g) Assume that Φ has a local linking at with respect to and . Then
We say that Φ satisfies the Cerami condition ((C) in short) if every sequence such that is bounded and as has a convergent subsequence. The following lemma derives from [, Proposition 3.2].
Lemma 2.1 
Let H be a Hilbert space, and are a family of functionals such that and are locally Lipschitz continuous. Assume that and satisfy (C). If there exists such that
Remark 2.1 The deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition . Therefore, if the (PS) condition is replaced by the (C) condition, (2.a)-(2.g) stated above still hold.
3 Compactness and critical group at infinity
In this section, we are going to prove the compactness of the associated energy functionals and to calculate the critical groups at infinity. First of all, let us introduce the variational structure for problem (BP).
3.1 Variational structure
The class E of functions such that , equipped with the inner product and norm as follows:
is linearly homeomorphic to . Denote . Throughout this paper, we always identify with .
Set , and
Then we can equivalently rewrite (BP) as a nonlinear algebraic system
Denote , , where I is the identity operator. Thus , . Set
then E has the decomposition . In the rest of this paper, the expression for always means , .
Define a functional by
where , . Then the Fréchet derivative of J at , denoted by , can be described by (see )
Remark 3.1 From (3.2) we see that is a critical point of J if and only if x is a solution of (3.1) (or equivalently (BP)). In addition, J is -differentiable with
3.2 Compactness of related functionals
Define a family of functionals , by
then the Fréchet derivative of at , denoted by , can be described by (see )
Lemma 3.1 Assume that (f∞) holds. For any sequences and , is bounded provided that
Moreover, for every , satisfies (C).
Proof Assume, for a contradiction, that is unbounded. Then there exists a subsequence, which we still call , with being nonempty such that
and either or, for any fixed , is a bounded sequence.
Noticing that [, Lemma 3.7], with its proof being modified slightly, is applicable here, we know that
Thus we have two cases to be considered.
Case 1. as . We have as and
By (f∞)(i), there exist and such that and for and . Then, for , and ,
Choose such that for and . It follows that
where . Since E possesses an equivalent norm defined by for , there exists a positive constant such that , . Thus, by (3.6) and (3.7),
where the equality holds because as for in case .
Case 2. as . In this case, by using (f∞)(ii), we can show that in the same way.
On the other hand, it follows from (3.5) that
which implies that
Note that , , it follows that . This contradiction proves the first conclusion.
By setting in the proven conclusion, we see that satisfies (C). The proof is complete. □
For , set , and . The following lemma is derived from [, Lemma 2.1].
Lemma 3.2 
If x is a solution of
then and hence it is also a solution of (3.1). Moreover, either or .
For , we say that () if () for .
Lemma 3.3 Let be the eigenvector corresponding to , , then can be chosen to satisfy . Moreover, for , neither nor .
Proof First we claim that or . Otherwise, by setting , we have
Since , it follows from (3.8) that
This contradiction proves the above claim. Thus can be assumed to satisfy and then . It follows by Lemma 3.2 that and the first conclusion holds. Further, for , and are orthogonal to each other, which implies that neither nor . The proof is complete. □
Lemma 3.4 Let the function be such that for . Assume that there exists such that
Then the functional
satisfies the (PS) condition, where .
Proof Let be such that
We only need to prove that is bounded. In fact, if is unbounded, there exists a subsequence, still called , such that as .
Let , then . There is a convergent subsequence of , call it again, such that as . For every , we have as , that is,
We claim that , since otherwise (3.11) leads to () for , which leads to , a contradiction. Thus we have by (3.9) that
which implies that there exists a subsequence of , still called , and , , such that
If , then as . Thus we can rewrite (3.11) as
Letting in (3.13) and using (3.12), we get
Since for , it follows from (3.14) that
which, by Lemma 3.2, implies that and hence . Thus, (3.14) can be rewritten as
Noticing that [, Lemma 3.4], with its proof being modified slightly, is applicable here, we know that w is an eigenvector corresponding to or . Since , it follows from Lemma 3.3 that . This contradiction completes the proof. □
3.3 Critical group at infinity
Lemma 3.5 Let f satisfy (f∞). Then
Proof We claim that there exists such that
otherwise there exist and such that and as , which contradict Lemma 3.1. Moreover, it is easy to see that . Thus, by Lemma 2.1, we have
On the other hand,
Note that is the unique critical point of with the Morse index and nullity . Then, by (2.b) and (2.e),
Similarly, we have . The proof is complete. □
4 Proofs of main results
Now we prove the main results of this paper. First, by applying (3.16) and (2.a), we know that J has a critical point satisfying
Define , . Then from (3.3) we know by calculation that is the solution space of the system , , where
Thus since B possesses non-degenerate order submatrixes. By (2.d)-(2.e), we further have
Proof of Theorem 1.1 First we give the proof for the case (i). By for , we know that is a strict local minimizer of J. Thus, by (2.f), we have correspondingly
Noticing that , by comparing (4.2) with (4.1), we have .
For , set for , for . Let . Then the critical points of
are exactly solutions of the problem
where , . By Lemma 3.4, we see that satisfies the (PS) condition. From the definition of and the assumption , , we know that there exists such that for . For any fixed with , define a function , . By the Lagrange mean value theorem, there exists such that
which implies that there exist and such that
In addition, let be the eigenvector of A corresponding to with , then (f∞), with , implies that
By Mountain Pass Theorem [17, 18], has a critical point with the critical group property for a mountain pass point , that is, . Noticing that satisfies (4.3), we get by Lemma 3.2 that and hence is also a mountain pass point of J, that is, .
The same argument shows that J has a nontrivial critical point with . Noticing that , by comparing the critical groups, we see that , and are three nontrivial critical points of J.
By the same argument as that for (4.1), we get , . If , and are all the nontrivial critical points of J, then and then (2.c) reads
a contradiction. Thus we claim that there exist at least four nontrivial critical points of J.
In the case (ii), we consider the functional −J. By applying (3.16) and (2.a), we know that −J possesses a critical point satisfying
Since for , is a strict local minimizer of −J and
Noticing that , we know by comparing (4.5) with (4.4) that . The rest of the arguments are similar to that in case (i) and will be omitted. The proof is complete. □
Proof of Theorem 1.2 In view of (3.3) and the assumption , , we see that is a non-degenerate critical point of J with the Morse index . Thus
Noticing that , we know by comparing (4.6) with (4.1) that .
We may assume that in (f0). For , set
where . Since as , there is a minimizer of . Thus
where for . By the definition of , the above equality can be rewritten as
From Lemma 3.4, we know that or . By assumption and , we know that θ is not a minimizer. Thus we have . In the same way as the proof of Lemma 3.2, we can prove that for . Thus is a local minimizer of J, therefore
Define , and consider the functional
where . A simple calculation shows that if z is a positive critical point of , then is a critical point of J, and, moreover, .
and its energy functional
where . By (f∞), we see that satisfies
It follows from Lemma 3.4 that satisfies the (PS) condition. If is not a strict local minimizer of J, then there exists infinitely many critical points near and the conclusion holds. Now we assume that is a strict local minimizer of J, then is a strict local minimizer of . In the same way as the proof of Theorem 1.1, we know that has a critical point , which is a mountain pass point of with and . Thus is also a critical point of with . Hence is a critical point of J with .
In a similar way, we know that J has a critical point with . Finally, by comparing the critical groups and by using the condition with , we see that , , and are four nontrivial critical points of J in which and are positive. The proof is complete. □
The proof of the following lemma is similar to that of [, Theorem 3.1] and is omitted.
Lemma 4.1 
Let f satisfy () (or ()). Then J has a local linking at with respect to the decomposition , where (or respectively).
Proof of Theorem 1.3 In view of (3.3) and the assumption , , we see that is a degenerate critical point of J with the Morse index and nullity . By Lemma 4.1 and (2.g), we have, corresponding to () or () respectively,
which, compared with (4.1), implies that in both of cases (i) and (ii). The rest of the proof is similar to that of Theorem 1.2 and will be omitted. The proof is complete. □
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The author is grateful for the referees’ careful reviewing and helpful comments. This work is supported by Beijing Municipal Commission of Education (KZ201310028031, KM2014).
The author declares that they have no competing interests.
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Zhang, X. Multiplicity results on discrete boundary value problems with double resonance via variational methods. Adv Differ Equ 2013, 309 (2013). https://doi.org/10.1186/1687-1847-2013-309
- discrete boundary value problem
- double resonance
- existence and multiplicity
- variational method