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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 102180, 6 pages

http://dx.doi.org/10.1155/2014/102180
Research Article

Existence of Positive Solutions of Lotka-Volterra Competition Model with Nonlinear Boundary Conditions

Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received 8 March 2014; Accepted 18 May 2014; Published 26 May 2014

Academic Editor: Dragos-Patru Covei

Copyright © 2014 Yang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Lotka-Volterra competition model with nonlinear boundary conditions is considered. First, by using upper and lower solutions method for nonlinear boundary problems, we investigate the existence of positive solutions in weak competition case. Next, we prove that , ; , ; , ; , , has no positive solution when one of the diffusion coefficients is sufficiently large.

1. Introduction

In this paper, we study the existence of positive solutions to the following problem with nonlinear boundary conditions:

Here,(H1) is an open bounded domain, and is the outward unit normal vector of the boundary ;(H2)for , , , , and are positive constants;(H3) and are strictly increasing functions in and ;(H4) .

In the usual interpretation of the competition model, and are population variables; it is natural to consider only nonnegative solutions of (1). There is clearly a trivial solution for all values of the parameters. In addition, for some values of parameters, there exist two semitrivial solutions and . More interesting are the so-called positive solutions or coexistence solutions, where both and are positive for all .

By using the positive operator theory, Ahn and Li [1] proved the existence of positive solutions to the following elliptic system: where , are functions in and , , , , and for ; and are increasing functions in and .

The main project of our paper is to investigate the existence of positive solutions to the problem (1). In Section 2 we state some known results, which are useful throughout this paper. Section 3 is devoted to proving the existence of positive solutions by using the upper and lower solutions method. When the diffusion coefficient or is sufficiently large, we will prove that problem (1) has no positive solution.

For the homogeneous Neumann boundary conditions, that is, and , problem (1) has been studied intensively by many authors. For the related results, please refer to, for instance, [28], [9, Section 4.3], and the references cited therein.

2. Preliminaries

In this section, we will introduce some notations and lemmas, which serve as the basic tools for the arguments to prove our results.

Throughout this paper, we will consider the solutions . For a given continuous function , let be the principal eigenvalue of the following eigenvalue problem: When the diffusion coefficient , we denote the first eigenvalue for (3) by .

The variational characterization of is We are concerned with the relation between the sign of and the function .

Lemma 1 (see [9]). The first eigenvalue of (3) has the following properties:

Lemma 2 (see [10]). Suppose that and is a positive constant satisfying . Then the following hold:(i) ;(ii) ;(iii) .

Next, we consider the nonlinear elliptic problem: where is strictly increasing with .

Definition 3. Let be global Lipschitz continuous in for all . The functions are called the upper and lower solutions of (6), if and satisfy

By using the upper and lower solutions method, the following result was obtained by Ahn and Li [1].

Lemma 4. Suppose that are upper and lower solutions of (6); then there exists a maximal solution of (6) such that .

Lemma 5 (see [1]). Let be positive constants and . Consider where , , and is strictly increasing. Then the following hold:(i)problem (8) has a solution for some , and where is dependent on ;(ii)if , then (8) has a unique positive solution.

Now we consider the following nonlinear boundary value problem:

Lemma 6. Let be an increasing and convex function in , and satisfy with . Assume that the function satisfies the following:(i) , is Lipschitz-continuous in , and the Lipschitz constant is independent of ;(ii) is decreasing in ;(iii) for , in for some constant .

If , then (10) has a unique positive solution. If , then is the only nonnegative solution of (10).

Proof. Let be the eigenfunction corresponding to the eigenvalue of problem (3); we can obtain when , due to the hypothesis (iii). With the assumptions on the function in mind, for a large , we have Therefore , where the operator is defined by the problem (8). is compact in the positive cone by Lemma 5. The function is monotone increasing on for sufficiently large . Therefore is increasing on the order interval with . Taking advantage of [1, Lemma 2.13], we know . The spectral radius by Lemma 2. Then, the result of [11, Theorem 7.6] ensures our conclusion.

Next we show the uniqueness. Suppose is a positive solution of (10). Let be a maximal solution of (10). We claim that .

Suppose . Then The first integral is positive, as is convex and . The last integral is nonpositive, since is decreasing in . This contradiction demonstrates . If , the proof is similar to [1, Lemma 2.17], and we omit it. This completes the proof.

By Lemma 5, we are able to conclude the following result.

Proposition 7. Suppose that are convex functions in . Then (1) has two semitrivial solutions and , for some , where is the unique positive solution of and is the unique positive solution of

Finally, we cite a strong maximum principle (see [5, Proposition 2.2]).

Lemma 8. Suppose that is smooth and . Assume that and satisfies If , then .

3. Existence and Nonexistence of Positive Solution

By using the upper-lower solutions argument for nonlinear boundary problems, we first study the existence of positive solutions to (1). Our method is technically and conceptually simple in the proof of existence results involving upper-lower solutions hypotheses and Leray-Schauder continuation argument. Next, we prove that problem (1) has no positive solution, if one of the diffusion coefficients is sufficiently large. Finally, we discuss the stability of semitrivial solutions.

We will show that the positive solutions of systems (1) have a priori bound.

Lemma 9. Suppose that is a positive solution of (1). Then and .

Proof. In view of the first equation of (1), satisfies As is continuous on the compact set and , thanks to Lemma 8, it is easy to know . In a similar manner, we obtain .

Next, we give the definitions of upper and lower solutions to (1).

Definition 10. Assume that . We called that and are the coupled upper and lower solutions of (1), if and satisfy

Theorem 11. Suppose that and are the coupled upper and lower solutions of (1) and . Then (1) has at least one solution and .

Proof. For any given and a sufficiently large positive constant , let Consider the following problem: Since , we see that (19) admits a unique solution by Lemma 5. Similarly, the problem has a unique solution . Denote , , and . We define the ordered interval and an operator by Thanks to Lemma 5, we have Hence, is bounded in .

We claim that is a compact operator. To see this, it suffices to prove that the operator is continuous. Suppose that in . Denote Then in . Let . By Lemma 5, we obtain Therefore, in . Note that is compact; it is deduced that in . It is obvious that is the solution of This shows that is continuous.

Now, we would like to prove . Suppose that and , where , . We first prove . In virtue of , we have that Let . Noting that and satisfies (19), it is obvious that On the contrary, we assume that is not true. By the strong maximum theory (see [12]), there exists , such that . Thanks to the Hopf boundary lemma, we know In view of (28), we get This is a contradiction with (29). Thus ; that is, . Similarly, , , and . By the Schauder fixed point theorem, has a fixed point in . The proof is complete.

Theorem 12. Suppose that and are convex functions in . Then the problem (1) has at least one positive solution.

Proof. By the assumption , there exists a constant such that

Let be the unique positive solutions of (13) and (14), respectively. Set , , where is sufficiently small. Then as and are positive on . To prove that and are the coupled upper and lower solutions of (1), it suffices to verify inequalities (17) in Definition 10. Consider the following.(i)Thanks to on , the following are obvious provided that is sufficiently small: (ii)By Lemma 8, we have that and on . In virtue of and , a direct computation gives

provided . Since the function is convex in , we know that, when , Similarly,

We have proved that are the coupled upper and lower solutions of (1). Taking advantage of Theorem 11, (1) has at least one positive solution. The proof is complete.

Next, we show that (1) has no positive solution, when the diffusion coefficient or is sufficiently large.

Theorem 13. There exists a positive constant such that when , problem (1) has no positive solution.

Proof. There exists a positive constant , independent of and , such that where Following the results of Lemma 9, we have Rewrite (1) as Note that . Multiplying the first equation of (39) by , and integrating on , we derive that, by Green’s identity, Holder’s inequality, and Poincare’s inequality, which implies that By Lemma 5, (36), and (41), we obtain Thanks to the Sobolev embedding theorem (see [12]), for , Review to the third equation of (1), we obtain In view of (43) and (44), it is easy to see Note that is increasing function and , as , and This is a contradiction. The same proof as above works equally well for the case when is large. This completes the proof.

Finally, we discuss the stability of semitrivial solutions.

Theorem 14. Suppose that are convex functions in . Then the following hold:(a)the semitrivial solution is unstable;(b)the semitrivial solution is unstable.

Proof. For part (a), to prove the stability of , we consider the corresponding elliptic system eigenvalue problem: First, observe that all eigenvalues of (47) are real since they are also eigenvalues of the second equation: Next, note that ; by Lemma 1, we have . Thus is unstable. In a similar way, we are able to conclude (b).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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