## Abstract

We investigate the existence and uniqueness of bounded, radially symmetric solutions in *R*^{3} of Δ*u*+*f*(*u*)=0, where *f*(*u*) is continuous, piecewise linear, and has three distinct real zeros. We give a detailed construction of the first two bound states which satisfy as *r*→∞. The first is the well known positive bound-state solution. The second solution changes sign and has exactly one zero in (0,∞). We prove that both of these solutions are unique.

## 1. Introduction

In 1972, Coffman (1972) proved the uniqueness of the positive, radially symmetric ground state solution in *R*^{3} of(1.1)That is, he showed that there is a unique solution of the initial-value problem(1.2)(1.3)such that(1.4)Here, . Condition (1.3) guarantees that the solution is bounded at *r*=0. To prove uniqueness of the positive ground state, Coffman followed an idea introduced by Kolodner (1955) and made use of the function which solves the equation of first variation. He developed a series of functionals and inequalities involving which determine the behaviour of the solution of (1.2) and (1.3) as *α* varies. He then used this information to prove the uniqueness of the solution of the full problem (1.2)–(1.4). McLeod & Serrin (1987) and McLeod (1993) studied more general equations such as(1.5)where the nonlinear term again has three zeros. They established the uniqueness of solutions of (1.5) and (1.4) over an extensive range of parameters *n*>1 and *p*>1. Kwong (1989) extended these results and proved the uniqueness of positive ground state solutions over the complete range *n*>1 and . The uniqueness of positive solutions in annular domains has been proved by Kwong & Zhang (1991) and Coffman (1996).

### (a) Sign changing bound states

In addition to positive ground states, there are bounded solutions of (1.5) which change sign any prescribed number of times on (0,∞), and satisfy(1.6)Such solutions are sometimes referred to as ‘higher bound states’ (e.g. see Soto-Crespo *et al*. 1991). Throughout this paper, the term *bound state* refers to any non-constant solution that satisfies (1.6). The existence of sign-changing bound-state solutions of (1.5) has been established by Coffman (1984), Jones & Kupper (1986) and McLeod *et al*. (1990). Later in this section, we discuss two applications in which this class of solutions plays an important role. Proving the uniqueness of sign-changing bound states has remained a challenging open problem. Here, it is natural to once again use to determine the behaviour of solutions as *α* varies. However, sign-changing solutions are no longer strictly monotone, and it is this property that has led to seemingly insurmountable technical difficulties in the analysis of . On this note, our goal is to develop methods that overcome these technical issues so that we may make use of to prove uniqueness of sign-changing bound-state solutions. Following Coffman, our approach is to focus on dimension *n*=3, and investigate a specific problem in which the nonlinearity has three zeros (figure 1). In particular, we study the existence and uniqueness of solutions of(1.7)(1.8)where(1.9)We let *u*(*r*,*α*) denote the solution of (1.7), such that . For ease of notation, the dependence of *u* on *α* is suppressed throughout most of the paper. Where appropriate, however, the full notation will be used. Since *f*(*u*) is bounded, standard arguments show that for each *α*∈*R*, the solution *u*(*r*,*α*) exists ∀*r*∈[0,∞). Also, because *f*(*u*) is odd, it follows that if *u*(*r*) is a solution of (1.7), then −*u*(*r*) is also a solution. Thus, we restrict our attention to *α*>0. In theorems 1.1 and 1.3, we prove the existence and uniqueness of the first two bound-state solutions (see figure 1). In order to obtain an understanding of how to use the function to prove uniqueness of bound-state solutions that change sign, a large part of this paper is devoted to studying the second bound state. To our knowledge, our proof of uniqueness of the second bound-state solution is the first of its kind. Future investigations will consider extensions of these results to bound states with more than one positive zero.

*(The positive bound state.) There is a unique α*_{0}*>*0*, such that if α=α _{0}, then*(1.10)

The proof of theorem 1.1 shows that .

*(The second bound state.) There is a unique α _{1}>*0

*, such that if α=α*0

_{1}, then there exists an r_{1}>*such that*(1.11)(1.12)

The proof of theorem 1.3 shows *α*_{1}>*α*_{0} and that .

### (b) Applications

We mention two applications in which sign-changing bound states play an important role. Soto-Crespo *et al*. (1991) investigated a partial differential equation (PDE) model of the self trapping of optical beams. The goal of these authors is to understand the mechanisms responsible for the breakup of optical beams into multi-bump structures called filaments. For this, they studied bounded, radially symmetric bound-state solutions that change sign. They linearized the PDE around these solutions and showed that they are unstable. Their analysis of the instability predicts that a small random perturbation from the symmetric bound-state solution will evolve into a multi-bump asymmetric solution as the variable corresponding to time increases. In addition, their methods predict the exact number of bumps in the resultant solution. Recently, Laing & Troy (2003) employed similar methods to study multi-bump formation in a fourth-order PDE model of a neural sheet. They showed that multi-bump axisymmetric bound-state solutions are unstable, and that small perturbations from these solutions evolve into multi-bump asymmetric solutions as time increases. Their analysis also predicts the exact number of bumps in the resultant asymmetric solutions.

### (c) Summary and outline

As stated above, an important goal of this paper is to understand the role of the equation of first variation in proving uniqueness of sign-changing bound-state solutions. By investigating the specific problem (1.7)–(1.9), it is hoped that a degree of insight will be provided that will lead to new uniqueness results for more general equations such as (1.5). The outline of the paper is as follows: preliminary results are given in §2, beginning with the form of (1.7) and its general solution in each of the regions described in (1.9). This is followed by the definition and basic properties of a functional that is used extensively in the proofs of the theorems. Section 3 gives the proof of theorem 1.1. In this section we also define and begin the analysis of the equation of first variation which *δ* solves. Section 4 gives the proof of theorem 1.3. Complete details are given for both the construction of the second bound state, as well as the use of *δ* to prove its uniqueness. Conclusions and directions for future study are given in §5. Appendix A contains proofs of technical lemmas from §4.

## 2. Preliminary results

There are three ODEs associated with (1.7)–(1.9). These are(2.1)(2.2)(2.3)The general solution of (2.1) is(2.4)The general solution of (2.2) is(2.5)and the general solution of (2.3) is(2.6)We will also make extensive use of the functional(2.7)which satisfies(2.8)The following property follows immediately from (1.7)–(1.9) and (2.7)–(2.8).

*If u*(*r*) *is a solution of* *(1.7)–(1.9)**, then Q*(*r*)*>*0 *∀r>*0.

An important consequence of lemma 2.1 is

*If u*(*r*) *is a solution of* *(1.7)–(1.9)* *then*(2.9)

Because of (2.7) and lemma 2.1, it follows that . The definition of *f*(*u*) in (1.9) shows that *f*(*u*)<0 if 0<*u*<1, hence *Q*(0)<0 when 0<*α*≤1. Therefore, it must be the case that *α*>1. An integration shows that(2.10)The right side of (2.10) is negative when and positive for . Thus, *Q*(0)>0 when and the lemma is proven. ▪

## 3. Proof of theorem 1.1

In this section we prove the existence of a unique positive bound-state solution of (1.7)–(1.9) (see figure 2). Because of lemma 2.2 we assume that(3.1)

Our goal is to prove

*There is a unique* *, such that if* *α=α*_{0} *then*(3.2)

The proof of theorem 3.1 makes use of several auxiliary lemmas beginning with

*If u*(*r*) *is a positive bound-state solution then u′*(*r*)*<*0 *∀r≥*0.

From (1.7)–(1.9) it follows that , since *f*(*α*)>0 when *α*>1. Thus *u*′(*r*)<0 on an interval (0,*ϵ*). If for some first , then . From (1.7)–(1.9) it follows that , hence . This, and lemma 2.1, contradict the assumption that *u* is a bound state. This completes the proof. ▪

Although we only need to consider , it is convenient to consider the entire range *α*>1. Substitution of (3.1) into (2.6) gives(3.3)for *r*>0 as long as Over the interval (0,*π*), the following result follows from (3.3), the definition of *Q*, and the fact that(3.4)

*Let α>*1. *Then*(3.5)(3.6)

To prove the existence of a unique, positive bound state we need to analyse the behaviour of solutions for all *α*>1. We begin with the set(3.7)The lemmas 3.4 and 3.5 determine the properties of *S*.

*If α∈S then u*(*r*) *is not a bound state. If* *then α∈S.*

Let *α*∈*S*. Then . Thus, (3.3) implies that ; hence *u*(*r*) cannot be a bound state by lemma 3.2. From (3.3) it follows that(3.8)Thus, *α*∈*S* if and the proof is complete. ▪

Below, lemma 3.5 shows that *S* is bounded above. The proof uses the fact that has a unique zero . This implies that(3.9)

*S is open, bounded and α _{*}≡*sup

*S<*4.

*If u*(

_{*}*r*)

*denotes the solution satisfying the initial conditions*

*, then*(3.10)

The definition of *S* and continuity of solutions with respect to initial conditions imply that *S* is an open interval of the form *S*=(1,*α*_{*}). If *u*(0)=4∈*S* then(3.11)However, substitution of *α*=4 and into (3.3) gives , contradicting (3.11), thus *α*_{*}<4. Finally, we analyse *u*_{*}(*r*). If , then continuity implies that *α*_{*}<sup *S*, contradicting the definition of *α*_{*}. Therefore, at a first . The definition of implies that . If , then (3.3) implies that , a contradiction. If , then (3.3) implies that and it follows from continuity that *α*_{*}>sup *S*, contradicting the definition of *α*_{*}. We conclude that and the lemma is proved. ▪

It follows from lemma 3.2 that *u*_{*}(*r*) is not a bound state, since . Thus, to find the positive bound state, we need to consider *α*>*α*_{*}. For such *α*, the next result shows that the solution must cross (see figure 2).

*Let α>α*_{*}. *There is a unique* *, such that*(3.12)*Furthermore, z*_{1} *is a decreasing, continuously differentiable function of α in* (*α*_{*},* ∞),*(3.13)(3.14)

Properties (3.13) and (3.14) will also be used in §4 where we analyse the bound state with one positive zero.

Let *α*>*α*_{*}. From (3.3) it follows that(3.15)for *r*∈(*π*,*r*_{*}), as long as From (3.15) we conclude that there is a unique *z*_{1}∈(*π*,*r*_{*}) satisfying (3.12), that *z*_{1} is continuously differentiable in *α*, and that the first two limits in (3.13) hold. From (3.3) and the fact that we obtain(3.16)Differentiation of (3.16) with respect to *α* results in(3.17)since *π*<*z*_{1}<*r*_{*} and *α*>*α*_{*}>1. Thus, exists and it follows from (3.16) and (3.17) that *z*_{1}(*α*)→*π*^{+} and as *α*→∞. Similarly, as . This completes the proof. ▪

To complete the proof of theorem 3.1, we need to analyse the behaviour of solutions when *r*>*z*_{1} and Here, *u*(*r*) solves (2.2) and has the form(3.18)To preserve continuity at *r*=*z*_{1}, we match the formulae for *u*(*z*_{1}) and *u*′(*z*_{1}) given in (3.3) with (3.18). Thus, we need to find *b*_{1} and *b*_{2} such that(3.19)(3.20)The solution of (3.19) and (3.20) is(3.21)(3.22)It follows from (3.18) that a bound state exists if *b*_{2}=0. Thus, we need to solve . It is easy to show that there is a unique such that . Since *z*_{1} is decreasing in *α*, there is a unique *α*_{0}>*α*_{*}, such that(3.23)From (3.22), (3.23) and (3.19) we conclude that(3.24)It is straightforward to obtain the improved estimates(3.25)We will use (3.24) and (3.25) in §4. Next, substitution of (3.23) into (3.21) gives ; hence (3.18) becomes(3.26)The solution defined in (3.26) satisfies as *r*→∞, as required for a bound state. Thus, *α*=*α*_{0} is the only value for which a positive bound state exists. We let *u*_{0}(*r*) denote the positive bound-state solution. Its full definition is(3.27)This completes the proof of theorem 3.1.

### (a) Equation of first variation

In §4 we prove the existence of the second bound state, a solution with exactly one positive zero. As was described in the introduction, to prove its uniqueness, we follow Coffman (1972) and use the function , which determines the behaviour of *u*(*r*) as *α* varies. The function *δ*(*r*) solves the equation of first variation, which is derived by taking the partial derivative of (1.7) with respect to *α*. The function *f*(*u*) defined in (1.9) is continuous on *R*, and has piecewise continuous, bounded first derivative. It follows from standard theory (e.g. see Hartman 1964, ch. V.) that *δ*(*r*) and *δ*′(*r*) are continuous on [0,∞), and *δ*″(*r*) is piecewise continuous. Because *f*′(*u*) is discontinuous at we will be careful to uniquely define *δ*(*r*) and *δ*′(*r*) to preserve continuity whenever On [0,*z*_{1}] we let *δ*(*r*) be denoted by *δ*_{0}(*r*). This function satisfies the equation of first variation, namely(3.28)(3.29)It follows from (3.3) that(3.30)Thus, for 0<*r*≤*z*_{1}, and(3.31)In §4 we use (3.30) and (3.31) to define *δ*(*r*) when *r*>*z*_{1} and

## 4. Proof of theorem 1.3

This section is devoted to proving that the second bound state exists and is unique (see figure 3). As was stated earlier, one of the main goals of this paper is to develop methods to circumvent the difficulties in using to prove uniqueness. Thus, as we construct the bound state, we simultaneously give all of the technical details of the construction and analysis of the behaviour of *δ*.

*There exists a unique α _{1}>*0

*, such that if α=α*

_{1}

*then there is a value*

*such that*(4.1)(4.2)

In the proof it will be shown that *α*_{1}>*α*_{0}, where *α*_{0} is the unique value at which the positive bound state exists.

For *r*>*z*_{1} the solution enters the region , where it has the form(4.3)Recall from (3.24) that *b*_{1}(*α*)>0 and . This, and (4.3), imply that there exist , such that (see figure 4)(4.4)By the implicit function theorem, *r*_{1} and *x*_{1} are continuously differentiable functions of *α* since *u*′(*r*_{1})<0 and *u*′(*x*_{1})<0. Also, it follows from (3.3) that as . This, and continuity, imply that(4.5)From (3.14) and continuity we also conclude that(4.6)Next, we extend the definition of to [*z*_{1},*x*_{1}] (see figure 4). Let *δ*_{1} denote *δ* over [*z*_{1},*x*_{1}] Since *u* satisfies when , then *δ*_{1} satisfies(4.7)At *r*=*z*_{1}, continuity requires that(4.8)where *δ*_{0}(*z*_{1}) and are both negative and are given in (3.31). Thus, an integration of (4.7) shows that . This implies that(4.9)A precise formula for *δ*_{1}(*r*) is obtained by solving (4.7) and (4.8). This results in(4.10)where(4.11)For future use we note that (3.25) together with (4.11) imply that(4.12)Also, the complete definition of *δ*(*r*) over [0,*x*_{1}] is(4.13)We now use *δ*_{1} in the derivation of two monotonicity properties which are needed later in the uniqueness proof. These are(4.14)First, we use the full notation *u*(*r*, *α*) and note that(4.15)Differentiating these equations with respect to *α* gives(4.16)since on [*z*_{1},*x*_{1}]. From (4.16), (4.4) and (4.9), it follows that d*r*_{1}/d*α*<0 and d*x*_{1}/d*α*<0 for all *α*>*α*_{0} and the proof of (4.14) is complete.

Next, as *r* increases past *x*_{1}, the solution enters the range and satisfies(4.17)From (4.4) and (4.17) it follows that for *r*>*x*_{1} as long as *u*(*r*)>−1. Thus, there is an *s*_{1}>*x*_{1} such that (see figure 5)(4.18)Furthermore, since *u*′(*x*_{1})→−∞ as *α*→∞, it follows from continuity that(4.19)Next, while the solution satisfies (4.17) and has the form(4.20)At *r*=*x*_{1} we have and (4.20) reduces(4.21)Requiring that the derivatives of (4.3) and (4.20) agree at *r*=*x*_{1} results in(4.22)Equations (4.21) and (4.22) can be uniquely solved for *A*_{1} and *A*_{2} since the matrix of coefficients of *A*_{1} and *A*_{2} has the determinant −1/*x*_{1}^{2}. Thus, the form of solution in (4.20) is uniquely defined for *r*>*x*_{1} as long as Setting *r*=*s*_{1} in (4.20) gives(4.23)since *u*(*s*_{1})=−1. Next, we let *t*_{1}=*s*_{1}+*π* and combine (4.20), (4.23) and the first derivative of (4.20) to obtain (see figure 5)(4.24)Since , it follows from (4.19) and (4.24) that(4.25)We will show below that there are three possibilities for the behaviour of *u*(*r*) when *r*>*t*_{1}. First, if *α*≫*α*_{0} then *u*=0 before *u*′=0 (figure 5, right panel). Second, if *α*−*α*_{0}>0 is small enough, then *Q*=0 before *u*=0. This implies that *u*′=0 before *u*=0 (figure 5, left panel). The third possibility is that the solution is a bound state with *u*′>0 and *u*<0 for all *r*>*t*_{1}, and (*u*, *u*′)→(0, 0) as *r*→∞. To prove these claims we define(4.26)It follows from the definition of *t*_{1}, (4.25) and continuity that if *α* is sufficiently large then there is an *r*_{2}>*t*_{1} such that and *u*(*r*_{2})=0 (figure 5, right panel). Thus, *α*∈*K*_{1} for *α*≫1. Since , then continuity implies that *K*_{1} is an open, semi-infinite interval. We claim that a bound state with exactly one positive zero exists for the value(4.27)To see this, let *u*_{1}(*r*) be the solution such that . There are two cases to consider.

*Case (i) α*_{1}>*α*_{0}. When *α*=*α*_{1}, the values *r*_{1}, *x*_{1}, *s*_{1} and *t*_{1} all exist. For reference, these points are illustrated in figures 4 and 5. If there is an , such that and −1<*u*_{1}(*r*)<0 for all , then (figure 5, right panel). It then follows from continuity that if *α*−*α*_{1}>0 is small enough, *u*(*r*) has exactly one zero on , and . Thus, *Q*(*r*)<0 on since *Q*′(*r*)≤0 for all *r*>0. However, there is an *r*_{2}(*α*)>*t*_{1}(*α*), such that *u*(*r*_{2}(*α*))=0 since *α*∈*K*_{1} when *α*−*α*_{1}>0, and at *r*_{2}(*α*) we have , a contradiction. We conclude that cannot exist. The second possibility when *α*=*α*_{1} is that there is an *r*_{2}(*α*_{1})>*t*_{1}(*α*_{1}), such that for all and *u*_{1}(*r*_{2}(*α*_{1}))=0. Uniqueness of solutions implies that . This, and continuity, imply that if *α*_{1}−*α*>0 is small enough then a first *r*_{2}(*α*)>*t*_{1}(*α*) exists with *u*(*r*_{2}(*α*))=0 and *u*′(*r*_{2}(*α*))>0. Therefore, *α*∈*K*_{1} if *α*_{1}−*α*>0 is sufficiently small, contradicting the definition of *α*_{1}. The remaining possibility is that and −1<*u*_{1}(*r*)<0 for all *r*>*t*_{1}. Standard theory implies that as *r*→∞, and therefore *u*_{1}(*r*) is a bound state with exactly one positive zero.

*Case (ii) α*_{1}*=α*_{0}. To eliminate this case we show that if *α*−*α*_{0}>0 is small enough then *r*_{2}(*α*) cannot exist, hence *α*∉*K*_{1}. This implies that *α*_{1}>*α*_{0}. We will use the following result, which is proved in appendix A.

*Let α>α*_{0}. *If* *then Q*(*t*_{1}(*α*))≤0.

Recall from (4.14) that *x*_{1}(*α*) is a decreasing, continuous function of *α* when *α*>*α*_{0}, that *x*_{1}(*α*)→∞ as , and *x*_{1}(*α*)→*π*^{+} as *α*→∞. Thus, there is an *α*^{*}>*α*_{0} such that *x*_{1}(*α*^{*})= and . We claim that *α*∉*K*_{1} if *α*∈(*α*_{0},*α*^{*}) Otherwise, there is an , such that , hence exists satisfying , and . Uniqueness of solutions implies that , and therefore(4.28)Since , then . However, lemma 4.3 holds because implies that , and therefore . Thus, , again since , contradicting (4.28). Therefore, cannot exist and we conclude that *α*∉*K*_{1} if . This eliminates the case *α*_{1}=*α*_{0}.

### (a) Uniqueness

To prove the uniqueness of the second bound state we use *δ*, the solution of the equation of first variation. The first step is to extend the definition of *δ*(*r*) to the range *r*>*x*_{1}, where (see figure 6).

When we let *δ*_{2}(*r*) denote *δ*(*r*). When the solution satisfies ; hence *δ*_{2}(*r*) satisfies(4.29)and has the form(4.30)At *r*=*x*_{1}, continuity requires that(4.31)Recall from (4.10) and (4.11) that(4.32)(4.33)Substituting (4.30) and (4.32) into (4.31), we obtain(4.34)(4.35)Solving these equations for *k*_{1} and *k*_{2} results in(4.36)(4.37)

Substituting *k*_{1} and *k*_{2} into (4.30), we obtain *δ*_{2}(*r*). It is possible that there are *α* values such that for all *r*>*x*_{1}. In this case, *δ*(*r*)=*δ*_{2}(*r*) for all *r*>*x*_{1}. However, it was proved above that when *α*−*α*_{0}>0 is large enough, then a first *r*_{2}>*x*_{1} exists at which *u*(*r*_{2})=0. This means that *u*(*r*) enters the region *u*> at a first *y*_{1}>*x*_{1}. In this case, we let *δ*_{3}(*r*) denote *δ*(*r*) in the range *r*>*y*_{1}, where <*u*(*r*)< When *u*(*r*)<, the solution satisfies ; hence *δ*_{3}(*r*) satisfies(4.38)and has the form(4.39)At *r*=*y*_{1}, continuity requires that *δ*_{3} satisfies(4.40)Using the function *δ*_{2}(*r*) derived above, and substituting (4.39) into (4.40), we obtain(4.41)(4.42)Solving these equations for *f*_{1} and *f*_{2} results in(4.43)(4.44)Substituting *f*_{1} and *f*_{2} into (4.39), we obtain *δ*_{3}. To prove uniqueness, we need to determine the signs of *δ*_{3}(*y*_{1}) and *f*_{2}. We do this in

*Let α>α*_{0} *such that y*_{1} *exists and π<x*_{1}*<* *Then*(4.45)

*If r*_{2}(*α*)*exists then*.*If u′*(*r*)*>*0*and**, then**. Furthermore,*

(4.46)

Substituting (4.36) and (4.37) into (4.43) and (4.44) leads to(4.47)(4.48)Recall from (4.12) that *p*_{1}>0 and *p*_{2}<0. This and (4.48) give(4.49)To determine the sign of *δ*_{3}(*y*_{1}), we combine (4.39), (4.47) and (4.48), and obtain(4.50)The first term in (4.50) is positive since *p*_{1}>0, *p*_{2}>0 and sin(*y*_{1}−*x*_{1})<0 when . To determine the sign of the second term we use the definitions of *p*_{1} and *p*_{2} given in (4.11) to obtain the expansion(4.51)The right side of (4.51) is negative since *x*_{1}>*z*_{1} and since (3.25) showed that . It then easily follows that the second term in (4.50) is positive and we conclude that *δ*_{3}(*y*_{1})<0. This completes the proof of (4.45). Next, to prove parts (i) and (ii) we consider any solution of (4.38) which satisfies *δ*_{3}(*y*_{1})>0 and has the form (4.39) where *f*_{2}>0. Then, *δ*_{3}(*r*)→∞ as *r*→∞ and it follows from a standard Sturm–Liouville-type argument applied to (4.38) that *δ*_{3} cannot change sign on [*y*_{1},∞). Thus, . This proves both (i) and (ii), since equation (4.38) is independent of *u*. This completes the proof of the lemma. ▪

We now give the final details of the proof of uniqueness. Let be any value where a bound state exists which has exactly one positive zero. Recall the notation *u*(*r*,*α*) that denotes the solution of (1.7) such that . Uniqueness is proved if we show that *u*(*r*,*α*) has at least two positive zeros . This implies that , where *α*_{1} is defined in (4.27). Since is a bound state with exactly one positive zero, then *z*_{1}, *r*_{1}, *x*_{1}, *s*_{1}, *t*_{1} and *y*_{1} all exist (see figure 7), and(4.52)It follows from (4.52) that *Q*(∞)=0, hence since *Q* is monotone decreasing. This and lemma 4.3 imply that . Next, we need to determine the relative location of in order to make use of lemma 4.4. For this we use the following technical result which is proved in appendix A.

*Let α>α*_{0}. *Suppose that π<x*_{1}(*α*)<, *that y*_{1}(*α*) *exists, and that* *, then*(4.53)

An important consequence of lemma 4.5 is that(4.54)Part (ii) of lemma 4.4 together with (4.54) imply that if then(4.55)We now use the same type of reasoning employed by Coffman (1972) in proving the uniqueness of the positive solution of (1.2). By (4.55) there is an , such that and . This and continuity imply that if and is sufficiently small, then *u*(*r*,*α*) has exactly one zero on , and(4.56)For such *α* we claim that *u*(*r*_{2}(*α*))=0 for some first . To see this, define . Then *v*(*r*) satisfies(4.57)for as long as <*u*(*r*,*α*)≤0. For small we have and . Using these values of *v* and *v*′ as initial conditions at , an integration of (4.57) shows that *v*′(*r*)>0 and for as long as *u*(*r*,*α*)<0. Thus, if it were the case that , then we would simultaneously have(4.58)a contradiction. Thus, it must be the case that *r*_{2}(*α*) exists and satisfies(4.59)We need to prove that *r*_{2}(*α*), this continues to exist for all . The first step is to observe that for *α*>*α*_{0}, as long as *r*_{2}(*α*) exists then *u*′(*r*_{2})>0 and . Thus, *Q*(*r*)>0 on the entire interval [0,*r*_{2}]. This and lemmas 4.3 and 4.5 imply that(4.60)for small . The same arguments show that inequalities (4.60) continue to hold for as long as *r*_{2}(*α*) exists. Next, differentiating (4.59) with respect to *α* gives(4.61)It then follows from (4.60), part (i) of lemma 4.4 and (4.61) that d*r*_{2}/d*α*<0 for small . Thus, *r*_{2}(*α*) moves to the left for small . Suppose that d*r*_{2}/d*α*=0 at some first . Then, exists and equation (4.61) reduces to(4.62)However, a repetition of the arguments given above shows that (4.60) is at . Thus, part (i) of lemma 4.4 implies that , contradicting (4.62). Therefore, *r*_{2}(*α*) exists and . This completes the uniqueness proof.

## 5. Conclusions and future directions

We have investigated the existence and uniqueness of the first two bound-state solutions of (1.7). The first solution is strictly positive, and the second solution has exactly one positive zero. To prove uniqueness of these solutions we made use of *δ*(*r*), the solution of the associated equation of first variation. The analysis of *δ* required that we precisely determine the location of points where solutions cross the lines *u*=± To accomplish this, we developed several technical lemmas and estimates. Our methods also apply when other piecewise linear functions are used.

A natural extension of this study is to prove the existence and uniqueness of bound-state solutions with more than one positive zero. For example, the third bound-state solution has exactly two positive zeros (figure 8). It is fairly easy to extend our methods to prove the existence of this solution, as well as the existence of solutions with any number of positive zeros. To successfully use *δ* to prove uniqueness of these bound states, it is again necessary to determine the location of points where solutions cross the lines *u*=± In principle, it appears that it is feasible, although technically challenging, to extend our techniques to locate these points.

As we mentioned earlier, it is hoped that the methods developed in this paper will provide a degree of insight for the development of new techniques to prove the uniqueness of multi-zero bound-state solutions of more general equations such as(5.1)which was described in §1.

## Acknowledgments

The author thanks Guido Sweers and the referees for making helpful suggestions. This work was supported in part by National Science Foundation grant DMS0412370.

## Appendix A

In this section, two results are proved that are used in the analysis of the second bound state. The first is lemma 4.3, which we restate as

*Let α>α*_{0}. *If x*_{1}(*α*)≥ *then Q*(*t*_{1}(*α*))*≤*0.

Suppose that the lemma is false. Then, there is an *α*>*α*_{0} such that(A1)Since , then . We prove below that this leads to(A2)The right side of (A 2) is negative at *x*_{1}=, and a straightforward analysis shows that it stays negative ∀*x*_{1}>. This contradicts (A 1). Thus, it remains only to derive (A 2). The first step is to show that(A3)The two lower bounds in (A 3) will be used to help complete the derivation of (A 2). The upper bound is used in the proof of lemma A 2. The first inequality in (A 3) follows from an algebraic computation since *x*_{1}≥. Next, we prove the second inequality. The mean value theorem implies that there is a *ζ*∈(*π*,*x*_{1}) such that(A4)Since when ≤*u*(*ζ*)≤1, then (A 4) and the definition of *Q* give(A5)Becauseis monotonic decreasing, then(A6)Recall that *u*(*r*) is decreasing on [*x*_{1},*s*_{1}] and that when −1≤*u*(*r*)≤− These facts together with (A 5) and (A 6) imply that(A7)An integration over [*x*_{1},*s*_{1}] gives the second inequality in (A 3). Next, we show that *s*_{1}−*x*_{1}<1. By the mean value theorem there is a *τ*∈(*x*_{1},*s*_{1}), such that(A8)Thus, if *s*_{1}−*x*_{1}≥1, then(A9)since when −1≤*u*≤− This contradicts the assumption that *Q*(*τ*)>0. We conclude that *s*_{1}−*x*_{1}<1 and the proof of (A 3) is complete. We now complete the derivation of (A 2). Since , then (A 5) extends to(A10)It is easily verified that *Q* satisfies the ODE(A11)Integrating (A 11) over [*x*_{1},*s*_{1}] gives . This can be written as(A12)since *u*(*s*_{1})=−1 and . Because of (4.24), we have(A13)From (A 12), (A 13) and the fact that , we obtain(A14)Substitution of (A 10) into (A 14) gives(A15)Finally, combining (A 15) and (A 3) gives (A 2). This completes the proof of the lemma. ▪

We now prove lemma 4.5, which is restated below as

*Let α>α*_{0}. *Suppose that π<x*_{1}(*α*)< *y*_{1}(*α*) *exists, and that* *. Then*(A16)

Again, suppose that the lemma is false. Then there is an *α*>*α*_{0}, such that *π*<*x*_{1}(*α*)<, *y*_{1}(*α*) exists , and(A17)To obtain a contradiction of (A 17), we need an upper bound on *s*_{1}(*α*)−*x*_{1}(*α*). The proof of lemma A 1 used the assumption the *Q* is positive on [0,*x*_{1}] to show that *s*_{1}−*x*_{1}<1. Below we improve this to *s*_{1}−*x*_{1}<0.57. In (A 9), also derived in the proof of lemma A 1 it was shown that *τ*∈(*x*_{1},*s*_{1}), such that(A18)Since *Q* is monotonically decreasing, it follows from (A 18) that *Q*(*s*_{1})≤*Q*(*τ*); hence(A19)since *u*(*s*_{1})=−1 and . Next, recall that . From this and (A 19) we conclude that(A20)Also, recall *s*_{1}<*x*_{1}+1, and that *x*_{1}< This reduces (A 20) to(A21)If it were the case that *s*_{1}−*x*_{1}≥0.57 then (A 21) further reduces to(A22)The right side of (A 22) is negative, contradicting our assumption that . We conclude that(A23)As will become clear below, in order to complete the proof of the lemma we also need to show that *y*_{1}−*t*_{1}<1. By the mean value theorem there is a , such that(A24)Note that , since . Thus, if *y*_{1}−*t*_{1}≥1 then(A25)This contradicts the assumption that . We conclude that *y*_{1}−*t*_{1}<1. Also, recall that *t*_{1}−*s*_{1}=*π*. These facts, together with (A 23), imply that(A26)This contradicts (A 17); hence the proof of the lemma is complete. ▪

## Footnotes

- Received September 20, 2004.
- Accepted March 23, 2005.

- © 2005 The Royal Society