Method of Lower and Upper Solutions for Elliptic Systems with Nonlinear Boundary Condition and Its Applications

Abstract

We develop the method of lower and upper solutions for a class of elliptic systems with nonlinear boundary conditions. As its application, an elliptic system modeling a population divided into juvenile and adult age groups is studied, and we find sufficient conditions in terms of the principal eigenvalue of the corresponding linearized system, to guarantee the existence of coexistence states of the above juvenile-adult model.

1. Introduction

Let be a bounded domain in with sufficiently smooth boundary . In this paper we study the elliptic systems of the form where is the unit exterior normal to , , , and is a linear differential operator of the form with symmetric coefficient matrix . We suppose that , , and for a certain . Moreover, is supposed to be strongly uniformly elliptic; that is, for some constant and every , .

System (1) with arises, in particular, in the study of steady state solutions of nonlinear parabolic equations of the form where is some second order, strongly uniformly elliptic differential operator. In this connection, nonlinear boundary conditions seem to be of particular importance. For the study of the stability of the solutions of the parabolic initial-boundary value problem (3), one has to have a good knowledge of the steady states, that is, of the solutions of (1) with . In the past few decades, the theory of monotone operators has been applied to boundary value problems of the form (3); see, for example, [1–3] and the references therein. In all of the above-mentioned papers, the boundary condition is of the special form , where is decreasing and is the conormal with respect to the differential operator . Besides these results, there are some scattered existence theorems for nonlinear Stecklov problems of the form where is supposed to be formally self-adjoint such that the homogeneous linear boundary value problem possesses a nontrivial solution; we refer the readers here to [4, 5] and the references therein. The stationary version of (3), which covers the above-mentioned several situations, has been studied by several authors; see Amann [6, 7] and Hess [8]. In particular, Amann [6] studied the stationary version of (3) and obtained a general existence theorem for it, namely, the result that the existence of a subsolution and a supersolution guarantees the existence of a solution. By transforming the stationary version of (3) into an equivalent fixed point equation in , he gave a new and more elegant proof for the above result.

Motivated by the above work, we will develop the method of lower and upper solutions for system (1) under the following assumptions:(H1)for , is -Hlder continuous in the first variable and locally Lipschitz in the second variable; is locally Lipschitz continuous;(H2)For , in , and satisfying on .

By a solution of (1) we mean a classical solution, that is, for each , such that for , and = for . For given and , we say if for every , and we define

Definition 1. Let . Then and are called ordered coupling upper and lower solutions of (1), if and they satisfy respectively.

The main result of this paper is the following.

Theorem 2. Assume that (H1) and (H2) are satisfied. Let and be ordered coupling upper and lower solutions of (1). Then (1) has at least one solution in .

Remark 3. Obviously, in the special case that , Theorem 2 generalize the corresponding Amann [6, Theorem 2.1]. In addition, we would like to point out that Theorem 2 also generalizes Wang [9, Theorem ], where the author developed the method of lower and upper solutions for the system (1) with linear boundary conditions.

The plan of the paper is as follows. In Section 2 we will develop the method of lower and upper solutions for elliptic system (1) and prove Theorem 2. Finally in Section 3, we will apply Theorem 2 to show the existence of positive solution of a juvenile-adult model.

2. Lower and Upper Solutions Method

Lemma 4. Let (H1) and (H2) hold. Then the system (1) is equivalent to the fixed point equation in . The map is completely continuous; that is, is continuous and maps bounded set into compact set.

Proof. For every , it is well known (cf. [10, 11] ) that the system has a unique solution as well as in , . Hence the Schauder estimates and the -estimates take the form respectively. Here and in the following denotes a positive constant (not necessarily the same in different formulas) which is independent of the functions appearing in these estimates. Hence (8) implies that is a bounded linear operator. Since and for every , Amann [6, Proposition 3.3] implies the existence of constant such that for every . Since the latter space is dense in , the estimate (10) implies that has a unique continuous extension for each , denoted again by , such that is also a bounded linear operator from to .
Now, let and be the Nemytskii operators generated by the vector fields and , respectively. Here and in the following we denote by an arbitrary, but fixed real number satisfying . This implies in particular that is compactly embedded into . We define by where denotes the usual trace operator. Then can be considered as a mapping of into . It is obvious that every solution of the system (1) is a fixed point of . Conversely, if is a fixed point of , then we can show that is also a solution of the system (1) by using the same methods as in the proof of Amann [6, Lemma 4.1].
Let . Then it is easy to see that and are bounded and continuous. Moreover, is a continuous linear operator such that . Since is also a bounded linear operator from to , and since is compactly embedded into for , it follows that is completely continuous from to .

Proof of Theorem 2. The regularity assumption (H1) for and implies the existence of positive constants such that Let Then the system (1) is equivalent to the system Moreover, and are also subsolution and supersolution for the system (17). Consequently, by Lemma 4, system (17) is equivalent to the fixed point equation in , and is completely continuous.
Now, let us show that .
It is clear that is a bounded closed convex subset of . Let and . Then and satisfy that On the other hand, since , we have and so , which together with Definition 1 yields that verifies Let . Then we can easily conclude from (12), (14), (18), and (19) that From the maximum principle for elliptic boundary value problems it follows that ; that is, . Similarly, by using (13) and (15), we can obtain . Consequently, maps into itself, and the existence of a fixed point follows from Schauder fixed point theorem.

3. Application to a Juvenile-Adult Model

In this section, we will apply the method of lower and upper solutions in Section 2 to study the existence of coexistence states of the following elliptic system describing two subpopulations of the same species competing for resources:

System (21) arises from population dynamics where it models the steady-state solutions of the corresponding nonlinear evolution problem [12], where and represent the concentrations of the adult and juvenile populations, respectively. The function gives the rate at which juveniles become adults and corresponds to the death rate of adult population. As adults give birth to juveniles, the function corresponds to the birth rate of the population. Juveniles are lost both through death and through becoming adults, the function corresponds to this overall loss. The Laplacian operator shows the diffusive character of and within . By using fixed point theory and lower and upper solutions method, several authors have studied the existence of coexistence states of the system (21), subject to Dirichlet or Neumann boundary conditions; see, for example, [13–16] and the references therein. We consider here the more general model (21), in which the boundary conditions may be interpreted as the conditions that the populations may pass through the boundary of the habitat. This is a mathematical model more closer to the reality.

In the rest of this section, we suppose that and satisfy (H2), the coefficients , and are positive functions in , and and are nonnegative functions defined on and satisfy (H1). We will use the notation if for . Apparently, since ,   for , the system is a linear cooperative system, for which we can give the following strong maximum principle.

Lemma 5. Let be such that and where equality does not hold in some of the equations in (23). Then (22) satisfies the strong maximum principle; that is, if such that then either (i) on or (ii) .

Proof. Otherwise, there exist not both identically zero satisfying (24) but not satisfying (ii) in the conclusion of the theorem. For , define and . Then there exists with such that for and either or has a zero in . Without loss of generality, we assume that there exists such that . Then by (23) and (24) we have That is, On the other hand, we can also deduce from (23) and (24) that and therefore the maximum principle for elliptic boundary value problems yields that or on . Since , we have that , and then from (26) we get . Since and are not both identically zero, . Thus, as and , we have, for , This is impossible since satisfy (23) and satisfy (24).

Corollary 6. Assume that and for all . Then the system (22) verifies the strong maximum principle.

Proof. We may get the desired results by using Theorem 9 and choosing , where is any positive constant.

Lemma 7. The cooperative system (22) has a principal eigenvalue; that is, there exists and such that and

Proof. Obviously, system (22) can be equivalently rewritten as where Let and . Then maps into . It is well known that if is sufficiently large, then is invertible such that is compact. Moreover, if is chosen sufficiently large such that and for all , then Corollary 6 yields that is strongly positive. It follows from the Krein-Rutman theorem that has a positive principal eigenvalue, which is denoted by . Therefore, there exists with such that . Consequently, and so has a principal eigenvalue .

Proposition 8. Assume that and are cooperative matrices. If , that is, for all and but , then .

Proof. By Lemma 7, there exists so that and . Then but , and from Lemma 5 we know that system satisfies the strong maximum principle. Hence if denotes the principal eigenvalue of the system , then it follows easily that . Clearly, has principal eigenvalue .

For each positive function , we denote by and the maximum and minimum of in , respectively. Let Then they are all positive constants. Applying Theorem 2, we will prove the following existence results of positive solutions of system (21).

Theorem 9. Suppose that satisfy If , then system (21) has at least one positive solution.

Proof. Assume . Then there exists with such that ; that is, Let and , where is given as in (33). In the following we will show that and satisfy the hypotheses of Theorem 2 if is defined as above and is chosen sufficiently small.
By the definition of , we have for all , and so In other words, we get Moreover, from (33) it follows that Similarly, we can deduce from (35) that
Let . Then, for any , we have and for all . Hence when and , we get when is sufficiently small; that is, Furthermore, since the function is nonnegative, we have that Applying the similar discussions as above, we can also prove that Consequently, it follows from Theorem 2 that system (21) has at least one positive solution in .

Remark 10. It is worth remarking that the results of Theorem 9 remain true if we use the weaker assumption and

Remark 11. In the special case on , the assumptions (34) and (35) are fulfilled. From Proposition 8 and similar argument used in the proof of [14, Theorem 3.2], we may show that the condition is also necessary for the existence of positive solutions of system (21). Consequently, if on , then system (21) has a positive solution if and only if . Hence Theorem 9 generalizes [14, Theorem 3.2] and [15, Corollary 18] in which only the problem with linear boundary conditions was considered.

Conflict of Interests

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

Acknowledgments

This work is supported by the NSFC (no. 11361054 and no. 11201378), SRFDP (no. 20126203110004), and Gansu provincial National Science Foundation of China (no. 1208RJZA258).