Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function
© Yulian An and Hua Luo. 2010
Received: 6 March 2010
Accepted: 11 August 2010
Published: 17 August 2010
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.
The existence and multiplicity of positive solutions for ordinary differential equations have been studied extensively in many literatures, see, for example, [1–3] 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. , Ouyang and Shi [7, 8], Shi , Korman and Ouyang [10, 11], Korman , Rynne , Bari and Rynne  (for th-order problems), as well as Korman and Li . 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.
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.
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.
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 .
Korman and Ouyang  obtained the unique positive solution curve of (1.3) under the condition (1.4). However they gave no information when can change sign. In , they did not treat the case that the equation contains a weight function.
On the other hand, suppose the following.
If , then we know from the proof in  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 ). 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 ).
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
has a nontrivial solution; otherwise, it is nondegenerate.
The proof is motivated by Lemma in .
Note that the right side of (2.11) is zero, which is a contradiction.
The following lemma is an important result in this paper.
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 .
Combining with (2.32), we obtain (2.26).
The following proof is motivated by the proof of Theorem in .
3. The Main Results and the Proofs
In this section we state our main results and proofs.
Definition 3.3 (see ).
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 .
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 .
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 .
The proof is similar to that of Proposition in , so we omit it.
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 , we have (i).
as a bifurcation problem from infinity. Note that (3.7) is also the same as to (1.1). The proof of Theorem in  ensures that (ii) is correct.
The following Lemma is an interesting and important result.
Now, we give the proof in two cases.
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 .
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 .
Our main result is the following.
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).
- (i)Since and , then . From Lemma 3.5(i) and the standard Crandall and Rabinowitz theorem on local bifurcation from simple eigenvalues , 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
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 , 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 .
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.
(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).
(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 , .
Theorems 3.12 and 3.13 extend the main result Theorem in , where for .
In this section, we give some examples.
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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