Research Article | Open Access
Global Structure of Positive Solutions of a Discrete Problem with Sign-Changing Weight
Let be an integer, . We are concerned with the global structure of positive solutions set of the discrete second-order boundary value problems ,, where is a parameter, changes its sign and .
Let be an integer, . In this paper, we are concerned with the global structure of positive solutions set of the discrete second-order boundary value problems where is a parameter, changes its sign, and for .
The boundary value problems with sign-changing weight arise from a selection-migration model in population genetics, see Fleming . changes sign corresponds to the fact that an allele holds an advantage over a rival allele at the same points and is at a disadvantage at others. The parameter corresponds to the reciprocal of the diffusion. So, the existence and multiplicity of positive solutions with sign-changing weight in continuous case has been studied by many authors, see, for example, [2–8] and the references therein.
For the discrete case, there are many literature dealing with difference equations similar to (1.1) subject to various boundary value conditions. We refer to [9–17] and the references therein. However, there are few papers to discuss the existence of positive solutions to (1.1) and (1.2) if changes sign on . Maybe the main reason is the spectrum of the following linear eigenvalue problems is not clear when changes its sign on .
In 2008, Shi and Yan  investigated the spectrum of the second-order boundary value problems of difference equations where is a symmetric and positive definite matrix. , , and satisfy(A1) changes its sign, and for .(A2), and , .
They proved that (1.4) has eigenvalues which have corresponding linearly independent and orthogonal eigenfunctions.
However, it's easy to see that (1.3) is not included in (1.4) under the condition (A2). Also, Shi and Yan  provided no information about the sign of the eigenvalues and no information about the corresponding eigenfunctions.
In this paper, we will show that (1.3) has two principal eigenvalues , and the corresponding eigenfunctions we denote by and don't change their sign on . Based on this result, using Rabinowitz's global bifurcation theorem , we will discuss the global structure of positive solutions set of (1.1) and (1.2).
The assumptions we are interested in this paper are as follows:(H1) with for ;(H2);(H3);(H3).
Our main result is the following.
Theorem 1.1. Assume that (H1), (H2) hold. (i)If (H3) holds, then there exist and such that (1.1), (1.2) has at least one positive solution for .(ii)If (H3) holds, then there exist and such that (1.1), (1.2) has at least one positive solution for .
The rest of the paper is arranged as follows. In Section 2, the existence of two principal eigenvalues of (1.3), and some properties of these two eigenvalues will be discussed. In Section 3, we will prove our main result.
2. Existence of Two Principal Eigenvalues to (1.3)
Recall that and . Let . Then is a Banach space under the norm . Let . Then is a Banach space under the norm .
It's well known that the operator , is a homomorphism.
Define the operator by
Theorem 2.1. Equation (1.3) has two principal eigenvalues and such that , and the corresponding eigenfunctions, we denoted by and do not change sign on .
Proof. Consider, for fixed , the eigenvalue problems
By Kelley and Peterson [22, Theorem 7.6], for fixed , (2.3) has -simple eigenvalues
and the corresponding eigenfunction has exactly simple generalized zeros.
Thus, is a principal eigenvalue of (1.3) if and only if .
On the other hand, let Clearly, is bounded below. Then .
For fixed , is an affine and so concave function. As the infimum of any collection of concave functions is concave, it follows that is a concave function. Also, by considering test functions such that and , it is easy to see that as . Thus, is an increasing function until it attains its maximum and is a decreasing function thereafter.
Since , must have exactly two zeros. Thus, (1.3) has exactly two principal eigenvalues, and , and the corresponding eigenfunctions we denoted by and don't change sign on .
Remark 2.2. From the proof of Theorem 2.1, it is not difficult to see that the following results hold.(i)If on and there exist at least one point such that , then (1.3) has only one principal eigenvalue .(ii)If on and there exist at least one point such that , then (1.3) has only one principal eigenvalue .
Now, we give some properties for the above principal eigenvalue(s).
Theorem 2.3. Let change its sign. Assume that there exists such that for . Then the followings hold. (i)If changes sign on , then , ;(ii)If , then .
Proof. For convenience, we only prove the case (i). It can be seen that for , , which implies and consequently, .
On the other hand, for , , which implies and consequently, .
Suppose that is a strict subset of , and denote the restriction of on . Consider the linear eigenvalue problems
Then we get the following result.
(i) If for , then (2.6) has only one positive principal eigenvalue such that .
(ii) If for , then (2.6) has only one principal eigenvalue such that .
(iii) If changes its sign on , then (2.6) has two principal eigenvalue and such that
Proof. Consider the following problems:
Let denote the th eigenvalue of (2.8), and the corresponding eigenfunction which has exactly generalized zeros in . Let be a Banach space under the norm . Let
We get . Similar to the proof of Theorem 2.1, we get the following assertions.(i)If for , then (2.6) has only one principal eigenvalue .(ii)If for , then (2.6) has only one negative principal eigenvalue .(iii)If changes its sign on , then (2.6) has two principal eigenvalue and .
Now, we prove that the inequalities in (i), (ii), and (iii) hold. For convenience, suppose that .
Let denote the extension of by zero on , that is, Then, which implies the desired results.
3. The Proof of the Main Result
First, we deal with the case .
Recall that , Let be such that Clearly, Let Then, is nondecreasing and Let us consider as a bifurcation problem from the trivial solution .
Equation (3.6) can be converted to the equivalent equation Further, we note that for near 0 in , since where
It is clear that any solution of (3.6) of the form yields a solution of (1.1) and (1.2). So, we focus on the shape of under the conditions (H1)–(H3) or (H1)–(H3), and we will show that crosses the hyperplane in .
Proof. Suppose on the contrary that there exists a sequence of positive solutions for (3.6) with and . Let , where . Then, by (H3), there exists such that implies .
Let and let and . Then we have for . Thus, On implies . Thus, Since for all , we have for all , and, thus, By the fact as , we get This contradiction completes the proof.
Lemma 3.2. Suppose that (H1)–(H3) hold. Then .
Proof. Assume on the contrary that , then, there exists a sequence such that for some positive constant independent of , since is unbounded. On the other hand, for all , since is the only solution of (3.6) for and . Meanwhile, satisfy By (H3), there exist a positive constant such that . Define a function by Then, . Let be the principal eigenvalue of linear eigenvalue problem Then, by Remark 2.2, . Subsequently, by Theorem 2.3, we know that This combine with Lemma 3.1, , which contradicts (3.15). Thus,
Lemma 3.3. Suppose that (H1), (H2), and (H3) hold. Then .
Proof. Let be such that as . Then,
If is bounded, say, , for some independent of , then we may assume that
Then, there exist two constants such that
Define two functions by
Let be the positive principal eigenvalue of the linear eigenvalue problems
Combining (3.22) and (3.24) with the relation using Theorem 2.3, we get This contradicts (3.22). So, is bounded uniformly for all .
Now, taking be such that We show that .
Suppose on the contrary that, choosing a subsequence and relabeling if necessary, for some constant . By (3.32), there exists such that and as . Thus,
Now, the proof can be divided into two cases.
Case 1 (). Consider the following linear eigenvalue problems By Theorem 2.4, (3.34) has a positive principal eigenvalue , and which contradicts (3.33).Case 2 (). Since is a solution of (3.7), we get This is a contradiction.
At last, we deal with the case .
Let us consider as a bifurcation problem from the trivial solution . The results of Rabinowitz  for (3.37) can be stated as follows: from /, there emanates an unbounded continua of positive solutions in .
It is clear that any solution of (3.37) of the form yields a solution of (1.1) and (1.2). Now, our proofs focus on the shape of . It will be proved that when (H1)–(H3) hold, then Proj/, and when (H1)–(H3) hold, Proj/, that is, crosses the hyperplane in . Since the proof is similar to the case , we omit it.
Remark 3.4. As an application of Theorem 1.1, let us consider nonlinear discrete problem with indefinite weight
where , with and , and
To find the principal eigenvalues of linear eigenvalue problem we rewrite (3.41) to the recursive sequence This together with the initial value condition imply that The last equation together with the boundary value condition imply that Thus by Theorem 1.1(i), (3.38), has a positive solution if .
The authors are grateful to the anonymous referee for their valuable suggestions. R. Ma is supported by NSFC (11061030).
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