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Open Access Research Article

Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

Yulian An1* and Hua Luo2

Author Affiliations

1 Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China

2 School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

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Boundary Value Problems 2010, 2010:207649  doi:10.1155/2010/207649


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2010/1/207649


Received:6 March 2010
Revisions received:18 July 2010
Accepted:11 August 2010
Published:17 August 2010

© 2010 The Author(s) Yulian An and Hua Luo.

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.

An exact multiplicity result of positive solutions for the boundary value problems , , , is achieved, where is a positive parameter. Here the function is and satisfies , for for some . Moreover, is asymptotically linear and can change sign only once. The weight function is and satisfies , for . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for lying in various intervals in . Moreover, we indicate how to extend the result to the general case.

1. Introduction

Consider the problem

(11)

where is a parameter and is a weight function.

The existence and multiplicity of positive solutions for ordinary differential equations have been studied extensively in many literatures, see, for example, [13] and references therein. Several different approaches, such as the Leray-Schauder theory, the fixed-point theory, the lower and upper solutions theory, and the shooting method etc has been applied in these literatures. In [4, 5], Ma and Thompson obtained the multiplicity results for a class of second-order two-point boundary value problems depending on a positive parameter by using bifurcation theory.

Exact multiplicity of positive solutions have been studied by many authors. See, for example, the papers by Korman et al. [6], Ouyang and Shi [7, 8], Shi [9], Korman and Ouyang [10, 11], Korman [12], Rynne [13], Bari and Rynne [14] (for th-order problems), as well as Korman and Li [15]. In these papers, bifurcation techniques are used. The basic method of proving their results can be divided into three steps: proving positivity of solutions of the linearized problems; studying the direction of bifurcation; showing uniqueness of solution curves.

Ouyang and Shi [7] obtained the curves of positive solutions for the semilinear problem

(12)

where is the unit ball in and . In [7], the following two cases were considered:

(i) does not change its sign on ; (ii) changes its sign only once on .

Korman and Ouyang [10] studied the problem

(13)

under the conditions and

(14)

They obtained a full description of the positive solution set of (1.3) and proved that all positive solutions of (1.3) lie on a single smooth solution curve bifurcating from the point and tending to in the plane. Condition (1.4) is very important to conclude the direction of bifurcation curve.

Of course a natural question is how about the structure of the positive solution set of (1.3) when changes its sign only once on ?

It is extremely difficult to answer such a question in general. So we shift our study to the problem (1.1) in this paper. We are interested in discussing the exact multiplicity of positive solutions of (1.1) with a weight function when changes its sign only once on .

Suppose the following.

(H1)One has with for some and for .

(H2) is concave convex that is, there exists such that

(15)

(H3)The limits and .

(H4) satisfies and if

In this paper, we obtain exactly two disjoint smooth curves of positive solutions of (1.1) under conditions (H1)–(H4). According to this, we can conclude the existence and exact numbers of positive solutions of (1.1) for lying in various intervals in .

Remark 1.1.

Korman and Ouyang [10] obtained the unique positive solution curve of (1.3) under the condition (1.4). However they gave no information when can change sign. In [7], they did not treat the case that the equation contains a weight function.

On the other hand, suppose the following.

()One has with . There exists such that and .

Remark 1.2.

If , then we know from the proof in [4] that the assumptions (H1) and (H3) imply that the component of positive solutions from the trivial solution and the component from infinity are coincident. However, these two components are disjoint under the assumptions (H1) and (H3) (see [5]). Hence, the essential role is played by the fact of whether possesses zeros in . In Section 3, we prove that (1.1) has exactly two positive solution curves which are disjoint and have no turning point on them (Theorem 3.8) under Conditions (H1)–(H4). And (1.1) has a unique positive solution curve with only one turning point (Theorem 3.9) if (H1) is replaced by (H1). The condition (H4) is used to prove the positivity of solutions of the linearized problems of (1.1) and the direction of bifurcation.

Our main tool is the following bifurcation theorem of Crandall and Rabinowitz.

Theorem 1.3 (see [16]).

Let and be Banach spaces. Let and let be a continuously differentiable mapping of an open neighborhood of into . Let the null-space be one dimensional and codim. Let . If is a complement of span in , then the solution of near forms a curve , where is a continuously differentiable function near and .

2. Notations and Preliminaries

Let with the norm

(21)

and let

(22)

with the norm

(23)

Set

(24)

equipped with the norm

(25)

Define the operator ,

(26)

Then, is a completely continuous operator.

Definition 2.1.

For a nontrivial solution of (1.1), is degenerate if the linearized problem

(27)

has a nontrivial solution; otherwise, it is nondegenerate.

Lemma 2.2.

Let (H1) and (H4) hold. For any degenerate positive solution of (1.1), the nontrivial solution of (2.7) can be chosen as positive.

Proof.

The proof is motivated by Lemma in [11].

Suppose to the contrary that has zeros on (0,1). Without loss of generality, suppose that . Note that and satisfy

(28)

(29)

respectively. We claim that has at most one zero in (0,1). Otherwise, let be the first two zeros of . Then,

(210)

Multiplying (2.9) by and (2.8) by , subtracting, and integrating over , we have

(211)

with to be specified. We denote the left side of (2.11) by and a constant by . Integrating by parts,

(212)

Let

(213)

on From (2.10), (2.13), and , we have

(214)

Note that the right side of (2.11) is zero, which is a contradiction.

Hence, has at most one zero in (0,1). Suppose that there is one point such that . Then,

(215)

Repeating the above proof on , we can get similar contradiction.

Finally, integrating the differential equation in (2.13), we can choose

(216)

In view of (H4), . So, the auxiliary function exists.

The following lemma is an important result in this paper.

Lemma 2.3.

Let (H1) and (H4) hold. Suppose that is a degenerate positive solution of (1.1). Then, the following are considered.

(i)All solutions of (1.1) near have the form for and some , where .

(ii)One has if is concave convex; if is convex concave.

Proof.

(i) The proof is standard. Let be such that . We will show that the conditions of Theorem 1.3 hold.

Since is a degenerate positive solution of (1.1), we denote the corresponding solution of (2.7) by . From Lemma 2.2 and the theory of compact disturbing of a Fredholm operator, is one dimensional and codim.

Now, we show that . Suppose to the contrary that . Then, there is a such that

(217)

(218)

Note that satisfies

(219)

(220)

Multiplying (2.17) by and (2.19) by , subtracting, and integrating on both sides, we obtain

(221)

However, the left side of (2.21) is equal to zero according to boundary conditions (2.18) and (2.20). This implies that . According to Theorem 1.3, the result (i) holds.

(ii) Substituting into (1.1), we obtain

(222)

Since then, by the implicit function theorem, the solution curve near is also Differentiating (2.22) twice with respect to , we have

(223)

Evaluating at , we obtain

(224)

Multiplying (2.24) by and (2.19) by , subtracting, and integrating, we get

(225)

According to (H1), (H4), and Lemma 2.2, we see that . Next, for the sign of , we consider the sign of .

We first prove that

(226)

Differentiating (1.1) and (2.19) with respect to , we have

(227)

(228)

Multiplying, (2.27) by and (2.28) by , subtracting, and integrating over , we get

(229)

with to be specified. Integrating by parts on the left side of (2.29),

(230)

Let

(231)

From (2.29), we get

(232)

Solving the equation , we can choose the auxiliary function

(233)

Combining with (2.32), we obtain (2.26).

The following proof is motivated by the proof of Theorem in [8].

Since , (2.26) implies that must change sign. If is concave convex, then there exists such that

(234)

Next, we claim that there exists , such that

(235)

Let . Then, , and . So, has at least one zero in . Moreover, we can prove that has only one zero in . Note that satisfies

(236)

We get

(237)

since and . Suppose that has more than one zero in (0, 1). Let be the last two zeros of , then we say that

(238)

We first prove the above statement. On the contrary, suppose that

(239)

Consider the problem

(240)

Obviously, is a subsolution and is a supersolution of (2.40), respectively. Note that . By the strong maximum principle, we obtain that on . This contradicts . Hence, the statement holds.

Now let us consider the claim related to . Multiplying (2.36) by and (2.37) by , subtracting, and integrating over , we get

(241)

since . Note that the left side is nonnegative. Such a contradiction implies that has only one zero in (0,1). By varying such that we can conclude the claim.

From the claim and , we have

(242)

Hence, from (2.25).

If is convex concave, then with a similar proof.

3. The Main Results and the Proofs

In this section we state our main results and proofs.

Definition 3.1.

Define

(31)

where is the first eigenvalue of the corresponding linear problem

(32)

Remark 3.2.

It is well known that the eigenvalues of (3.2) are given by

(33)

For each , algebraic multiplicity of is equal to 1, and the corresponding eigenfunction has exactly simple zeros in .

Definition 3.3 (see [7]).

Let . Then is said to be superlinear (resp., sublinear) on if (resp., ) on . And is said to be sup-sub (resp., sub-sup) on if there exists such that is superlinear (resp., sublinear) on , and superlinear (resp., sublinear) on .

Lemma 3.4.

(i) Let and (H4) hold. Suppose that is a point where a bifurcation from the trivial solutions occurs and that is the corresponding positive solution bifurcation curve of (1.1). If there exists such that is superlinear (resp., sublinear) on , then tends to the left (resp., the right) near .

(ii) Let and (H4) hold. Suppose that is a point where a bifurcation from infinity occurs and that is the corresponding positive solution bifurcation curve of (1.1). If there exists such that is superlinear (resp., sublinear) on and (resp., ) for , then tends to the right (resp., the left) near .

Proof.

The proof is similar to that of Proposition in [7], so we omit it.

Lemma 3.5.

Let (H1)–(H4) hold, let be a bounded and closed interval, and let . Suppose that are positive solutions of (1.1). Then,

(i), if ,

(ii), if .

Proof.

Let be such that

(34)

Clearly,

(35)

Let us consider

(36)

as a bifurcation problem from . Note that (3.6) is the same as to (1.1). From Remark 3.2 and the standard bifurcation theorem from simple eigenvalues [17], we have (i).

Let us consider

(37)

as a bifurcation problem from infinity. Note that (3.7) is also the same as to (1.1). The proof of Theorem in [5] ensures that (ii) is correct.

Lemma 3.6.

Let (H1), (H4) hold. Suppose that is a positive solution of (1.1). Then,

(38)

Proof.

Suppose to the contrary that

(39)

By (1.1) and (H1), we have . Note that . By the uniqueness of solutions of initial value problem, the problem

(310)

has a unique solution . This contradicts .

The following Lemma is an interesting and important result.

Lemma 3.7.

Let (H1)–(H4) hold. Suppose that is a positive solution of (1.1), then is nondegenerate.

Proof.

From conditions (H1)–(H3), we can check easily that

(311)

In fact, let , then

(312)

since . Note that , if and , if . This together with (3.12) implies that and (3.11).

Now, we give the proof in two cases.

Case I ().

On the contrary, suppose that is a degenerate solution with , then for all. By (3.11), we get

(313)

Multiplying (1.1) by and (2.7) by , subtracting, and integrating, we have

(314)

By Lemma 2.2, (3.13), and for all, the right side of (3.14) is negative. This is a contradiction.

Case II ().

On the contrary, suppose that is a degenerate solution with . According to Lemmas 2.2 and 2.3, we know that all solutions of (1.1) near satisfy for and some , where . It follows that for close to we have two solutions and with strictly increasing in and with strictly decreasing in . We will show that the lower branch is strictly increasing for all .

Note that for close to and all . Let be the largest where this inequality is violated; that is, and for some . Differentiating (1.1) with respect to ,

(315)

We can extend evenly , and on , then we obtain

(316)

By the strong maximum principle, we conclude that for all . This contradicts that .

By Lemma 2.3, we get at every degenerate positive solution. Hence, there is no degenerate positive solution on the lower branch . However, the lower branch has no place to go. In fact, there must exist some positive constant such that for any lying on . Hence, the lower branch cannot go to the axis. And it also cannot go to the axis, since (1.1) has only the trivial solution at .

So, is nondegenerate.

Our main result is the following.

Theorem 3.8.

Let (H1)–(H4) hold. Then the following are considered.

(i)All positive solutions of (1.1) lie on two continuous curves and without intersection. bifurcates from to infinity and ; bifurcates from to infinity and . There is no degenerate positive solution on these curves. For any , , and for any , .

(ii)Equation (1.1) has no positive solution for has exactly one positive solution for but and has exactly two positive solutions for (see Figure 1).

Proof.

(i) Since and , then . From Lemma 3.5(i) and the standard Crandall and Rabinowitz theorem on local bifurcation from simple eigenvalues [17], is the unique point where a bifurcation from the trivial solution occurs. Moreover, by Lemma 3.4, the curve bifurcates to the right. We denote this local curve by and continue to the right as long as it is possible. Meanwhile, by Lemma 3.6, there is no positive solution of (1.1) which has the maximum value on . So, if , then . From (1.1), we have

(317)

where . Obviously, there exists a constant such that if is bounded. Hence, cannot blow up.

On the other hand, Lemma 3.7 and the implicit function theorem ensure that cannot stop at a finite point .

From the above discussion, we see that can be extended continuously to infinity and . Meanwhile, the maximum values of all positive solutions of (1.1) are less than .

Now, we consider positive solutions of (1.1), for which the maximum value on is greater than .

Let us return to consider (3.6) as the bifurcation problem from infinity. Note that (3.6) is also the same as to (1.1). Since by Theorem and Corollary in [18], there exists a subcontinuum of positive solutions of (3.6) which meets . Take as an interval such that and as a neighborhood of whose projection on lies in and whose projection on is bounded away from . Then, there exists a neighborhood such that any positive solution of (1.1) satisfies for and some and at , where denotes the normalized eigenvector of (3.2) corresponding to . So,

Hence, is a continuous curve, and we denote it by . It tends to the right from Lemma 3.4(ii). From Lemma 3.7 and the implicit function theorem, can be continued to a maximal interval of definition over the axis. We claim that cannot blow up if is bounded. In fact, suppose that there exists a positive solutions sequence of (1.1) and such that as . Then, by Lemma 3.5(ii), . This is a contradiction. On the other hand, the implicit function theorem implies that cannot stop at a finite point . Thus, and if .

Finally, we show that both curves and are the only two positive solutions curves of (1.1). On the contrary, suppose that is a positive solution of (1.1) with . Without loss of generality, assume that . Note that is nondegenerate, so we can extend it to form a curve. We denote this curve by and the corresponding maximal interval of definition by . Since all positive solutions of (1.1) are nondegenerate, according to the implicit function theorem, we must have that

(318)

It follows that from Lemma 3.5(ii). But all solutions near can be parameterized by for and some ; thus, . This contradicts that .

Similarly, we can show that every positive solution of (1.1), the maximum value on of which is less than lies on .

(ii) The result (ii) is a corollary of (i).

Next, we will give directly other theorems. Their proofs are similar to that of Theorem 3.8. So, we omit them.

Theorem 3.9.

Let and (H2)–(H4) hold. Then, the following are considered.

(i)All positive solutions of (1.1) lie on a single continuous curve . And bifurcates from to the right to a unique degenerate positive solution of (1.1), then it tends to the left to .

(ii)Equation (1.1) has no positive solution for , and has exactly one positive solution for , and has exactly two positive solutions for (see Figure 2).

Remark 3.10.

In fact, if we reverse the inequalities in (H1), (H1), (H2), we will obtain corresponding results similar to Theorems 3.8 and 3.9.

Also using the method in this paper, we can obtain the exact numbers of positive solutions for the Dirichlet problem

(319)

where is a parameter. We assume that

(H) with , and for all.

Definition 3.11 .

Define

(320)

where is the first eigenvalue of the corresponding linear problem of (3.19).

Theorem 3.12.

Let (H1), (H2), (H3), and (H4) hold. Then, the following are considered.

(i)All positive solutions of (3.19) lie on a single continuous curve . And bifurcates from to the right to a unique degenerate positive solution of (3.19), then it tends to the left to .

(ii)Equation (1.1) has no positive solution for but has exactly one positive solution for and has exactly two positive solutions for .

Theorem 3.13.

Let (H1), (H2), (H3), (H4) hold. Then

(i)All positive solutions of (3.19) lie on two continuous curves and without intersection. bifurcates from to infinity and ; bifurcates from to infinity and . There is no degenerate positive solution on these curves. For any , , and for any , .

(ii)Equation (3.19) has no positive solution for , and has exactly one positive solution for , and has exactly two positive solutions for .

Remark 3.14.

Theorems 3.12 and 3.13 extend the main result Theorem in [10], where for .

4. Examples

In this section, we give some examples.

Example 4.1.

Let

(41)

Then, satisfies (H1), (H2), and (H3). Moreover, , , , and .

Example 4.2.

Let

(42)

Then, satisfies (H), (H2), and (H3). Moreover, and .

Example 4.3.

Let . Here, , for all, and is a large enough constant. Then, satisfies (H4). On the other hand, functions which satisfy (H) can be found easily.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. An is supported by SRFDP (no. 20060736001), YJ2009-16 A06/1020K096019, 11YZ225. Luo is supported by grant no. L09DJY065.

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