Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Abstract

Let be an integer and . We show the existence of the principal eigenvalues of linear periodic eigenvalue problem , and we determine the sign of the corresponding eigenfunctions, where is a parameter, and in , and the weight function changes its sign in . As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.

1. Introduction

In 1997, Constantin [1] studied the following linear periodic eigenvalue problem: where , , and . He obtained that if changes its sign, then (1) and (2) have infinite real eigenvalues, , such that Equation (1) with plays a crucial role in the study of the water shallow equation; see [2–5].

Let be an integer and . In 2005, Wang and Shi [6] discussed the eigenvalues of a discrete periodic boundary value problem where , , and are real functions with for , for , and , and is the spectral parameter. They showed the existence of eigenvalues of (4) and (5) and calculated the numbers of eigenvalues.

Ji and Yang [7] considered a class of boundary value problems of the second-order difference equation (4) with the more general boundary conditionsThe class of problems considered include those with antiperiodic, Dirichlet, and periodic boundary conditions. They focused on the structure of eigenvalues and comparisons of all eigenvalues of (4) and (6), as the coefficients , , and change their signs. They got a very interesting result: the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function.

Gao and Ma [8] studied the eigenvalues of periodic and antiperiodic eigenvalue problems of discrete linear second-order difference equation (4) with sign-changing weight. They find that these two problems have real eigenvalues (including the multiplicity), respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the certain order relation.

However, all of above papers provide no information about the sign of the eigenfunctions of (4) and (5). In particular, they give no information about the sign of the eigenfunctions corresponding to two simple eigenvalues and .

It is the purpose of this paper to show the existence of the principal eigenvalues and determine the sign of the corresponding eigenfunctions for linear periodic eigenvalue problem where and in , and the weight function changes its sign in . Our approach is motivated by Brown and Lin [9] and Smoller [10], where the infimum of Rayleigh quotient is used to characterize the principal eigenvalues for diverse linear eigenvalue problems of elliptic equations with infinite weight.

For the other recent results on the spectrum structure of discrete linear eigenvalue problems with one-sign weight, see Sun and Shi [11], Shi and Chen [12], Jirari [13], Bohner [14], and Agarwal et al. [15] and the references therein.

The rest of the paper is organized as follows. In Section 2, we show the existence of the principal eigenvalues of (7) and determine the sign of the corresponding eigenfunctions. In Section 3, we apply our spectrum theory and the well-known Rabinowitz bifurcation theorem to show the existence of positive solutions for nonlinear discrete periodic boundary value problemunder , where and in , is a continuous function, and changes its sign.

2. Existence of Principal Eigenvalues

In this section, we consider the linear eigenvalue problem (7) under the following assumptions:

(H0) , and for some .

(H1) , and there exists a proper subset, , of , such that for and for . Let be the number of elements in . Then is the number of elements in .

The difference operator is , defined by where

Define a linear operator byThen, it is easy to see that is isomorphism. Moreover, is a self-adjoint operator whose spectrum consists only of the real eigenvalues.

Let us denote the norm and the inner product of by respectively. For , we have from [16, Lemma ] that Thus, we may define a functional

To study the principal eigenvalues of (7), we need the following preliminary lemmas.

Lemma 1. Let (H0) and (H1) hold. Assume there is a nonnegative eigenfunction corresponding to an eigenvalue of (7). Then, for all .

Proof. Suppose that is a nonnegative eigenfunction corresponding to the eigenvalue . Then, is a nonnegative eigenfunction corresponding to the eigenvalue of Letand . Then, is strongly positive if is large enough. Clearly, and are the eigenvalue and eigenfunction of if and only if and are the corresponding eigenvalue and eigenfunction of the problem Since is a self-adjoint operator, its spectrum contains only real eigenvalues: Moreover, by the well-known Krein-Rutman theorem [17, Theorem ], is simple, and its corresponding eigenfunction is of one sign.
So, the eigenvalue is simple and the corresponding eigenfunction does not change sign in . Notice and are eigenvalue and the corresponding eigenfunction of (16) and (17), respectively, and is not orthogonal to . This together with the fact that eigenfunctions corresponding to distinct eigenvalues are orthogonal implies that must be an eigenfunction of corresponding to . Hence, . By the spectral theorem that is, for all .

Lemma 2. Let (H0) and (H1) hold. Let Then, .

Proof. By the spectral theorem, for all , where is the first eigenvalue of . Note that (H0) implies that Hence, if with , then Hence

Lemma 3. If , then is not an eigenvalue of (7) possessing a nonnegative eigenfunction.

Proof. If , there exists such that and ; that is, and so . The required result is now an immediate consequence of Lemma 1.

Lemma 4. Assume (H1) and . Then, there exists ( depends on ) such that for all .

Proof. Let , where .
We claim that In fact, for , we have from the fact thatthat

Lemma 5. Let (H0) and (H1) hold. LetThen, .

Proof. If with , then it follows from the fact thatthatHence,

Lemma 6. If , then is not an eigenvalue of (7) possessing a nonnegative eigenfunction.

Proof. If , then there exists such that and ; that is, and so . The required result is now an immediate consequence of Lemma 1.

Lemma 7. Assume (H1) and . Then, there exists ( depends on ) such that for all .

Proof. Let , where .
For , we have from that

Theorem 8. If (H0) and (H1) hold, then (7) has exactly two principal eigenvalues and , such that

(1) ;

(2) the algebraic multiplicity of and is ;

(3) the eigenfunctions and corresponding to the eigenvalues and are of one sign.

Proof. Consider the linear eigenvalue problem It is easy to see that is an eigenvalue for (7) with corresponding eigenfunction if and only if is an eigenvalue of , and, accordingly, is an eigenvalue of (38) with corresponding eigenfunction . The least eigenvalue of is given by Since for all , . Because of how we defined , there exists a sequence such that and Therefore, and so . Hence, is the least eigenvalue of (38) and so is simple and the corresponding eigenfunction can be chosen to be positive on .
Using the same method, with obvious changes, we may prove the algebraic multiplicity of is and the eigenfunction corresponding to the eigenvalue is of one sign.

3. Existence of Positive Solutions

As an application, we consider the existence of positive solutions of the discrete nonlinear problem (8) and (9).

In this section, we assume that

(H2) for all ;

(H3) for all , where and for all .

Let be the principal eigenvalue of and let be an eigenfunction corresponding to .

Theorem 9. Assume that (H0)–(H3) hold. Then

(1) (8) and (9) have at least one positive solution for ;

(2) (8) and (9) have at least one positive solution for .

Proof. Step  1. A bifurcation result.
We extend the function to a continuous function defined on by setting, for , Obviously, within the context of positive solutions, problem (8) and (9) is equivalent to the same problem with replaced by . Furthermore, is an odd function for . In the sequel of the proof, we shall replace with . However, for the sake of simplicity, the modified function will still be denoted by .
Recall that whereLet be such that where Let us consider as a bifurcation problem from the trivial solution . A solution of (48) is a pair which satisfies (48). It is easy to see that any solution of (48) of the form yields a solution of (8) and (9).
Equation (48) can be converted into the equivalent equationFurther, we note that , since where and is given by [18, Theorem ]. We denote (49) by Clearly, and are completely continuous andwith respect to varying in bounded intervals. Notice that if is the eigenvalue of , then it also is the characteristic value of .
We say that a solution of (51) is nontrivial if there exists such that . Denote by the closure in of the set of all nontrivial solutions of (51) with .
Theorem  1.3 in [19] yields the existence of a maximal closed connected set in such that and at least one of the following conditions holds:
(i) is unbounded in .
(ii) There exists a characteristic value of , with , such that .
Step  2. In what follows, we prove several properties which will eventually lead to the fact that condition (ii) above does not hold.
Claim  1. Suppose , and then is a characteristic value of . Let be a sequence of nontrivial solutions of (51), converging to in .
Setting, for all , , we have As is bounded in and is completely continuous, there exist and a subsequence of , which we denote in the same way, such that Hence, we conclude by (53) that Therefore, we have with and in particular . Accordingly, is a characteristic value of .
Claim  2. There exists such that .
By contradiction, we can suppose that there exists a sequence of nontrivial solutions of (51), converging in to some . Arguing as in the proof of Claim  1, we set , and we have and conclude that, possibly passing to a subsequence, in , which contradicts .
Claim  3. if and only if .
This follows from the fact that , and hence , is odd with respect to the second variable.
In the sequel, we denote by the positive cone in ; that is, and we denote by its interior and by its boundary.
Claim  4. There exists a neighborhood of in such that, for all , either , or , or .
It is an immediate consequence of the fact that for all and the well-known Crandall-Rabinowitz local bifurcation theorem; see Crandall and Rabinowitz [20] and Kielhöfer [21].
Claim  5. Assume and . Suppose further that is the limit of a sequence in , with for all . Then, .
We first show that .
Suppose, by contradiction, that . Then, we can take such that Hence, we get As satisfies for some , let be the Green function for the linear boundary value problem Since the Green function for (62) satisfies on (see [18, Theorems  2.1 and 2.2] for details), this yields for , contradicting . Therefore, we conclude that . We next show that . By Claim  1, is a characteristic value of . Setting and arguing as in the proof of Claim  1, we conclude that, possibly passing to a subsequence, in , where is an eigenfunction of (7) associated with . Since , we conclude that .
Claim  6. For all , either , or , or .
Set By Claim  4, and, subsequently, is a closed subset of .
Let us verify that is open in . Suppose this is not the case. Then, there exist and a sequence in converging to . We may assume that for all ; hence, by Claim  5, we obtain , contradicting the fact that . As is connected and , we conclude that .
By Claim  6 we have that if , then and hence condition (ii) above does not hold. Consequently, condition (i) is valid.
Step  3. Next, let us show that joins to .
Let satisfy We note that for all , since is the only solution of (51) for and .
We first show that there exists a constant , such that Suppose (67) does not hold, then choosing a subsequence and relabelling if necessary, it follows that We divide the equation by and set , and we have for all . Since is bounded in , after taking a subsequence and relabelling if necessary, we have that for all , where with .
It follows from (66) and (68) that either Combining this fact with (H3) and where with , we have this is a contradiction, and therefore (67) holds.
According to (67), we have Since solve we divide (75) by and set . Then, we get Since is bounded in , after taking a subsequence and relabelling if necessary, we have that for some with . Moreover, by (72) and (74), we can show that Therefore, where , again choosing a subsequence and relabelling if necessary.
We claim that Suppose on the contrary that . Since is a solution of (78) it follows that . By the openness of , we know that there exists a neighborhood such that which contradicts the facts that in and . Therefore, . Moreover, by Sturm-Liouville eigenvalue theory, . Therefore, joins to .
Therefore, crosses the hyperplane in , and, accordingly, (8) and (9) have at least one positive solution.

Remark 10. Let us consider the nonlinear problemwhereObviously,with and . Thus On the other hand, it follows from (26) and the fact that Therefore, Theorem 9 (2) yields the existence of at least one positive solution of (81).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Ma and Xu completed the main study together and Ma wrote the manuscript; Long was responsible for checking the proofs process and verified the calculation. Moreover, all the authors read and approved the final version of the manuscript.

Acknowledgments

This work was supported by NSFC (no. 11671322) and NSFC (no. 11361054).