Multiplicity results on discrete boundary value problems with double resonance via variational methods
© Zhang; licensee Springer. 2013
Received: 15 May 2013
Accepted: 14 October 2013
Published: 8 November 2013
The existence of solutions for a class of difference equations with double resonance is studied via variational methods, and multiplicity results are derived.
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].
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
which has completely the same form as its counterpart in . However, instead of (), in this paper we assume 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 .
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 .
() 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).
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.
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
is called the q th critical group of Φ at , where .
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.
The following results are due to Su .
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 
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
is linearly homeomorphic to . Denote . Throughout this paper, we always identify with .
then E has the decomposition . In the rest of this paper, the expression for always means , .
3.2 Compactness of related functionals
Moreover, for every , satisfies (C).
and either or, for any fixed , is a bounded sequence.
Thus we have two cases to be considered.
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.
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 
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 .
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. □
satisfies the (PS) condition, where .
We only need to prove that is bounded. In fact, if is unbounded, there exists a subsequence, still called , such that 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
Similarly, we have . The proof is complete. □
4 Proofs of main results
Noticing that , by comparing (4.2) with (4.1), we have .
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.
a contradiction. Thus we claim that there exist at least four nontrivial critical points of J.
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. □
Noticing that , we know by comparing (4.6) with (4.1) that .
where . A simple calculation shows that if z is a positive critical point of , then is a critical point of J, and, moreover, .
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).
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. □
The author is grateful for the referees’ careful reviewing and helpful comments. This work is supported by Beijing Municipal Commission of Education (KZ201310028031, KM2014).
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