1. Introduction

The nonlinear boundary value problem where is the Laplacian of is the diffusion coefficient, and are positive constants, and is a smooth bounded region with in . This model describes phosphorus cycling in stratified lakes (see [1]). In particular, it illustrates the decrease in the amount of phosphorus in the epilimnion (upper layer) and the rapid recycling that occurs when the hypolimnion (lower layer) is depleted of oxygen. Here, is the mass or concentration of phosphorous (P) in the water column, and is the rate of P input from the watershed. The rate of recycling of P is given by , where is the maximum recycling rate. The assumption here is that the recycling is primarily from the sediments. The same equation has also been used to describe plant colonization of barren soils in drylands (see [2]). In this case, is the amount of barren soil, and represents erosion by wind and runoff.

It is known that when with , a positive solution exists for all , and this solution is unique if is increasing. The existence of multiple positive solutions to such problems has also been studied extensively (see [3–6]). On the other hand, proving multiplicity results for nonlinearities with a falling zero (say at ) is very challenging and often remains an open problem (see [7] and example (iv) in [4]). For such problems, the solution space is restricted as (see Figure 1). Our model falls in this category.

Figure 1: Restricted solution space for the falling zero problem.

Instead of working with the particular reaction term in (1.1), we will prove our results for a class of functions which satisfy the following hypothesis: on and for .

To state our multiplicity result, for , let where is the largest inscribed ball on , (see Figure 2) and is the unique positive solution of in on ,.

Figure 2: Definition of and .

Now, we establish the following result.

Theorem 1.1. Let be such that is nondecreasing in . Assume that there exists and such that , then (1.1) has three positive solutions for .

We will use the method of subsupersolutions to prove our results. By a subsolution (supersolution) of (1.1), we mean a function such that on and for every . Then, the following lemma holds.

Lemma 1.2. Let be a subsolution of (1.1), and let be a supersolution of (1.1) such that , then, (1.1) has a solution such that .

To establish our main multiplicity result (Theorem 1.1), we use the following very useful result discussed in [8, 9].

Lemma 1.3. Suppose that there exists a subsolution , a strict supersolution , a strict subsolution , and a supersolution for (1.1) such that , , and let , then, (1.1) has at least three distinct solutions , and such that .

Note here that by we mean that and .

We prove Theorem 1.1 in Section 2. The proof of Theorem 1.1 is motivated by the arguments in [7] where the authors establish a multiplicity result for a model used to describe a logistically growing species with grazing. In Section 3, we analyze in detail the phosphorus cycling model when has a unique positive zero . This will be the case when and . We will prove that an -shaped bifurcation curve occurs when and . This analysis turned out to be quite nontrivial and challenging. This study is motivated by the results in [10, 11] where such a multiplicity result for the case was discussed. Here, we extend this study for the higher dimension case. We also obtained more detailed analytical and computational results for the case , which are presented in the appendix.

2. Proof of Theorem 1.1

To establish the multiplicity result, we have to construct a subsolution , a strict supersolution , a strict subsolution , and a supersolution for (1.1) such that , , and . Clearly, is a strict subsolution since . For the large supersolution, choose where . Then, making a positive supersolution.

Now for the smaller strict supersolution, define . Since in . Here, . Hence, is a strict supersolution.

We will now construct the strict subsolution . Let where is defined so that the function is strictly increasing on and (see Figure 3).

Figure 3: Graph of .

Let Note that and . Now define and where is the solution of and is the largest inscribed ball in . Then, and on . We will now establish that on . Then, on , while outside , we have , and hence, will be a strict subsolution.

First, we will show that . Now But . Hence, we get But . Hence, .

Next, to establish on , we will show that on . This will be sufficient, since . Now and in the interval , and hence, in that interval. For , we have We also know that . Hence, if . But , and this minimum is achieved at . Since , we can choose and such that . Hence, on . This implies . Thus, is a strict subsolution of (1.1) if . Furthermore, , that is, . Moreover, can be chosen large enough so that and . Hence, by Lemma 1.3, Theorem 1.1 holds.

3. Results for the Example

First, we will analyze some properties of this nonlinearity. We will show that for large we can find values of for which the function satisfies , and we will also identify and such that is increasing in . Clearly, , , and .

Proposition 3.1. If , then there exists two points such that and for .

Proof. We have . So , when . Let , and let . We have , , and . Since , we can see that achieves a maximum of at . If , then the line will cut at exactly two different points. Hence, if , then there are exactly two positive points such that for .

Proposition 3.2. If , then there exists a unique such that .

Proof. From Figure 5, we can see that if , then has a unique positive zero. Since , we obtain . So, . Hence, if . On analyzing , we see that the positive inflection of occurs at . Thus, , and hence ensures that there exists a unique such that .

Choose and . Thus, given , we can find large so that is increasing on , and there exists a unique such that , that is, satisfies . Now, we will prove that the other assumptions in Theorem 1.1 hold in the given example.

We will select and such that . The point at which the function has a minimum would be an ideal choice for .

Proposition 3.3. If , then has the shape given in Figure 6.

Proof. We have . Hence, the critical points of are given by , and in particular, the nonzero critical points are given by . Solving for , we get the positive critical points as and . Note that if , then and are positive real roots of with . Hence, has a relative maximum at and a relative minimum at .

Since as and , we have . Furthermore, it is clear from Figure 4 that as , so for . Thus, we have for large and we choose . We also choose such that (see Figure 7).

Figure 4: and .

Figure 5: Graph of with and .

Figure 6: Graph of .

Figure 7: Graph of .

The following estimates hold for and for .

Proposition 3.4. For ,  ,  .

Proof. (i) By the shape of the graph of established in Propositions 3.1 and 3.3 (see Figure 5), it is enough if we prove that and . We have , and for . Thus, for large .
(ii) We have , simplifying which we get . Define . If , then , and if , then . We have , and for . Hence, for large . Note that since and as , for .

Now we will discuss the existence of at least three positive solutions for a certain range of (see Theorem 1.1). Our aim is to prove that for and , . It is enough if we prove that Note that for and . Also, . Applying the estimates we obtained for , and to the above inequality, we get the following: Simplifying the above, we can see that if Clearly, this inequality is true for ; hence, Theorem 1.1 holds.

Appendix

Using the quadrature method (see [12]), it follows that (A.1) has a positive solution if and only if where and . Further, is symmetric about and is given by

Equation (A.2) describes the bifurcation curve of positive solutions of (A.1), and it follows by results in [12] that and . Furthermore, from [4], we have where . Note that and . Hence, if there exists a point such that , (see Figure 8) then for a certain range of ; thus, the bifurcation diagram must be at least -shaped. We will now prove that such a exists when and .

Figure 8: Graph of .

Consider the case . Clearly, given , then for satisfies (G1)-(G2) (see Proposition 3.3). Hence, is defined for all .

We have . Clearly, (see Proposition 3.3); choose . Thus, , and hence, if .

We finally used Mathematica to compute in the case when and plotted the bifurcation diagrams. We found that the bifurcation diagrams are, in fact, exactly -shaped when multiplicity occurred. Figures 9 and 10 describe the bifurcation diagrams for a certain value of and .

Figure 9: Bifurcation diagram with and . Here, .

Figure 10: Bifurcation diagram with and . Here, .