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
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, [1–3] and references therein. Several different approaches, such as the LeraySchauder theory, the fixedpoint 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 secondorder twopoint 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 thorder 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
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
under the conditions and
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
(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 nullspace 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
and let
with the norm
Set
equipped with the norm
Define the operator ,
Then, is a completely continuous operator.
Definition 2.1.
For a nontrivial solution of (1.1), is degenerate if the linearized problem
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
respectively. We claim that has at most one zero in (0,1). Otherwise, let be the first two zeros of . Then,
Multiplying (2.9) by and (2.8) by , subtracting, and integrating over , we have
with to be specified. We denote the left side of (2.11) by and a constant by . Integrating by parts,
Let
on From (2.10), (2.13), and , we have
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,
Repeating the above proof on , we can get similar contradiction.
Finally, integrating the differential equation in (2.13), we can choose
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
Note that satisfies
Multiplying (2.17) by and (2.19) by , subtracting, and integrating on both sides, we obtain
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
Since then, by the implicit function theorem, the solution curve near is also Differentiating (2.22) twice with respect to , we have
Evaluating at , we obtain
Multiplying (2.24) by and (2.19) by , subtracting, and integrating, we get
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
Differentiating (1.1) and (2.19) with respect to , we have
Multiplying, (2.27) by and (2.28) by , subtracting, and integrating over , we get
with to be specified. Integrating by parts on the left side of (2.29),
Let
From (2.29), we get
Solving the equation , we can choose the auxiliary function
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
Next, we claim that there exists , such that
Let . Then, , and . So, has at least one zero in . Moreover, we can prove that has only one zero in . Note that satisfies
We get
since and . Suppose that has more than one zero in (0, 1). Let be the last two zeros of , then we say that
We first prove the above statement. On the contrary, suppose that
Consider the problem
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
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
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
where is the first eigenvalue of the corresponding linear problem
Remark 3.2.
It is well known that the eigenvalues of (3.2) are given by
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 supsub (resp., subsup) 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
Clearly,
Let us consider
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
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,
Proof.
Suppose to the contrary that
By (1.1) and (H1), we have . Note that . By the uniqueness of solutions of initial value problem, the problem
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
In fact, let , then
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
Multiplying (1.1) by and (2.7) by , subtracting, and integrating, we have
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 ,
We can extend evenly , and on , then we obtain
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
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
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
where is a parameter. We assume that
(H) with , and for all.
Definition 3.11 .
Define
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
Then, satisfies (H1), (H2), and (H3). Moreover, , , , and .
Example 4.2.
Let
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), YJ200916 A06/1020K096019, 11YZ225. Luo is supported by grant no. L09DJY065.
References

Ambrosetti, A, Hess, P: Positive solutions of asymptotically linear elliptic eigenvalue problems. Journal of Mathematical Analysis and Applications. 73(2), 411–422 (1980). Publisher Full Text

Asakawa, H: Nonresonant singular twopoint boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 44(6), 791–809 (2001). PubMed Abstract  Publisher Full Text

Erbe, LH, Wang, H: On the existence of positive solutions of ordinary differential equations. Proceedings of the American Mathematical Society. 120(3), 743–748 (1994). Publisher Full Text

Ma, R, Thompson, B: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Analysis: Theory, Methods & Applications. 59(5), 707–718 (2004). PubMed Abstract  Publisher Full Text

Ma, R, Thompson, B: Multiplicity results for secondorder twopoint boundary value problems with nonlinearities across several eigenvalues. Applied Mathematics Letters. 18(5), 587–595 (2005). Publisher Full Text

Korman, P, Li, Y, Ouyang, T: An exact multiplicity result for a class of semilinear equations. Communications in Partial Differential Equations. 22(34), 661–684 (1997). Publisher Full Text

Ouyang, T, Shi, J: Exact multiplicity of positive solutions for a class of semilinear problem. II. Journal of Differential Equations. 158(1), 94–151 (1999). Publisher Full Text

Ouyang, T, Shi, J: Exact multiplicity of positive solutions for a class of semilinear problems. Journal of Differential Equations. 146(1), 121–156 (1998). Publisher Full Text

Shi, J: Exact multiplicity of solutions to superlinear and sublinear problems. Nonlinear Analysis: Theory, Methods & Applications. 50(5), 665–687 (2002). PubMed Abstract  Publisher Full Text

Korman, P, Ouyang, T: Exact multiplicity results for two classes of boundary value problems. Differential and Integral Equations. 6(6), 1507–1517 (1993)

Korman, P, Ouyang, T: Solution curves for two classes of boundaryvalue problems. Nonlinear Analysis: Theory, Methods & Applications. 27(9), 1031–1047 (1996). PubMed Abstract  Publisher Full Text

Korman, P: Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems. Journal of Differential Equations. 244(10), 2602–2613 (2008). Publisher Full Text

Rynne, B: Global bifurcation for 2mth order boundary value problems and infinitely many solutions of superlinear problems. Journal of Differential Equations. 188(2), 461–472 (2003). Publisher Full Text

Bari, R, Rynne, B: Solution curves and exact multiplicity results for 2mth order boundary value problems. Journal of Mathematical Analysis and Applications. 292(1), 17–22 (2004). Publisher Full Text

Korman, P, Li, Y: On the exactness of an Sshaped bifurcation curve. Proceedings of the American Mathematical Society. 127(4), 1011–1020 (1999). Publisher Full Text

Crandall, MG, Rabinowitz, PH: Bifurcation, perturbation of simple eigenvalues and linearized stability. Archive for Rational Mechanics and Analysis. 52, 161–180 (1973)

Crandall, MG, Rabinowitz, PH: Bifurcation from simple eigenvalues. Journal of Functional Analysis. 8, 321–340 (1971). Publisher Full Text

Rabinowitz, PH: On bifurcation from infinity. Journal of Differential Equations. 14, 462–475 (1973). Publisher Full Text