- Research Article
- Open access
- Published:
Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function
Boundary Value Problems volume 2010, Article number: 207649 (2010)
Abstract
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ1_HTML.gif)
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 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ3_HTML.gif)
under the conditions and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ5_HTML.gif)
(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.
(H1′) 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ6_HTML.gif)
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ7_HTML.gif)
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ8_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ9_HTML.gif)
equipped with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ10_HTML.gif)
Define the operator ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ11_HTML.gif)
Then, is a completely continuous operator.
Definition 2.1.
For a nontrivial solution of (1.1), is degenerate if the linearized problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ14_HTML.gif)
respectively. We claim that has at most one zero in (0,1). Otherwise, let
be the first two zeros of
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ15_HTML.gif)
Multiplying (2.9) by and (2.8) by
, subtracting, and integrating over
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ16_HTML.gif)
with to be specified. We denote the left side of (2.11) by
and a constant
by
. Integrating by parts,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ17_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ18_HTML.gif)
on From (2.10), (2.13), and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ19_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ20_HTML.gif)
Repeating the above proof on , we can get similar contradiction.
Finally, integrating the differential equation in (2.13), we can choose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ23_HTML.gif)
Note that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ25_HTML.gif)
Multiplying (2.17) by and (2.19) by
, subtracting, and integrating on both sides, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ26_HTML.gif)
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
(2.22)
Since then, by the implicit function theorem, the solution curve near
is also
Differentiating (2.22) twice with respect to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ28_HTML.gif)
Evaluating at , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ29_HTML.gif)
Multiplying (2.24) by and (2.19) by
, subtracting, and integrating, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ31_HTML.gif)
Differentiating (1.1) and (2.19) with respect to , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ33_HTML.gif)
Multiplying, (2.27) by and (2.28) by
, subtracting, and integrating over
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ34_HTML.gif)
with to be specified. Integrating by parts on the left side of (2.29),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ35_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ36_HTML.gif)
From (2.29), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ37_HTML.gif)
Solving the equation , we can choose the auxiliary function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ39_HTML.gif)
Next, we claim that there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ40_HTML.gif)
Let . Then,
, and
. So,
has at least one zero in
. Moreover, we can prove that
has only one zero in
. Note that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ41_HTML.gif)
We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ42_HTML.gif)
since and
. Suppose that
has more than one zero in (0, 1). Let
be the last two zeros of
, then we say that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ43_HTML.gif)
We first prove the above statement. On the contrary, suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ44_HTML.gif)
Consider the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ48_HTML.gif)
where is the first eigenvalue of the corresponding linear problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ49_HTML.gif)
Remark 3.2.
It is well known that the eigenvalues of (3.2) are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ50_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ51_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ52_HTML.gif)
Let us consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ53_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ54_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ55_HTML.gif)
Proof.
Suppose to the contrary that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ56_HTML.gif)
By (1.1) and (H1), we have . Note that
. By the uniqueness of solutions of initial value problem, the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ58_HTML.gif)
In fact, let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ60_HTML.gif)
Multiplying (1.1) by and (2.7) by
, subtracting, and integrating, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ61_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ62_HTML.gif)
We can extend evenly , and
on
, then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ63_HTML.gif)
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).
![figure 1](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Fig1_HTML.jpg)
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
(3.17)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ65_HTML.gif)
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).
![figure 2](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Fig2_HTML.jpg)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ66_HTML.gif)
where is a parameter. We assume that
(H) with
, and
for all
.
Definition 3.11 .
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ67_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ68_HTML.gif)
Then, satisfies (H1), (H2), and (H3). Moreover,
,
,
, and
.
Example 4.2.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F207649/MediaObjects/13661_2010_Article_903_Equ69_HTML.gif)
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.
References
Ambrosetti A, Hess P: Positive solutions of asymptotically linear elliptic eigenvalue problems. Journal of Mathematical Analysis and Applications 1980,73(2):411-422. 10.1016/0022-247X(80)90287-5
Asakawa H: Nonresonant singular two-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2001,44(6):791-809. 10.1016/S0362-546X(99)00308-9
Erbe LH, Wang H: On the existence of positive solutions of ordinary differential equations. Proceedings of the American Mathematical Society 1994,120(3):743-748. 10.1090/S0002-9939-1994-1204373-9
Ma R, Thompson B: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Analysis: Theory, Methods & Applications 2004,59(5):707-718.
Ma R, Thompson B: Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues. Applied Mathematics Letters 2005,18(5):587-595. 10.1016/j.aml.2004.09.011
Korman P, Li Y, Ouyang T: An exact multiplicity result for a class of semilinear equations. Communications in Partial Differential Equations 1997,22(3-4):661-684. 10.1080/03605309708821278
Ouyang T, Shi J: Exact multiplicity of positive solutions for a class of semilinear problem. II. Journal of Differential Equations 1999,158(1):94-151. 10.1016/S0022-0396(99)80020-5
Ouyang T, Shi J: Exact multiplicity of positive solutions for a class of semilinear problems. Journal of Differential Equations 1998,146(1):121-156. 10.1006/jdeq.1998.3414
Shi J: Exact multiplicity of solutions to superlinear and sublinear problems. Nonlinear Analysis: Theory, Methods & Applications 2002,50(5):665-687. 10.1016/S0362-546X(01)00775-1
Korman P, Ouyang T: Exact multiplicity results for two classes of boundary value problems. Differential and Integral Equations 1993,6(6):1507-1517.
Korman P, Ouyang T: Solution curves for two classes of boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications 1996,27(9):1031-1047. 10.1016/0362-546X(95)00108-8
Korman P: Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems. Journal of Differential Equations 2008,244(10):2602-2613. 10.1016/j.jde.2008.02.014
Rynne B: Global bifurcation for 2mth order boundary value problems and infinitely many solutions of superlinear problems. Journal of Differential Equations 2003,188(2):461-472. 10.1016/S0022-0396(02)00146-8
Bari R, Rynne B: Solution curves and exact multiplicity results for 2mth order boundary value problems. Journal of Mathematical Analysis and Applications 2004,292(1):17-22. 10.1016/j.jmaa.2003.08.043
Korman P, Li Y: On the exactness of an S-shaped bifurcation curve. Proceedings of the American Mathematical Society 1999,127(4):1011-1020. 10.1090/S0002-9939-99-04928-X
Crandall MG, Rabinowitz PH: Bifurcation, perturbation of simple eigenvalues and linearized stability. Archive for Rational Mechanics and Analysis 1973, 52: 161-180.
Crandall MG, Rabinowitz PH: Bifurcation from simple eigenvalues. Journal of Functional Analysis 1971, 8: 321-340. 10.1016/0022-1236(71)90015-2
Rabinowitz PH: On bifurcation from infinity. Journal of Differential Equations 1973, 14: 462-475. 10.1016/0022-0396(73)90061-2
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
An, Y., Luo, H. Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function. Bound Value Probl 2010, 207649 (2010). https://doi.org/10.1155/2010/207649
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/207649