- Research Article
- Open access
- Published:
Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem
Boundary Value Problems volume 2010, Article number: 458015 (2010)
Abstract
We are concerned with the existence of positive solutions of singular second-order boundary value problem ,
,
, which is not necessarily linearizable. Here, nonlinearity
is allowed to have singularities at
. The proof of our main result is based upon topological degree theory and global bifurcation techniques.
1. Introduction
Existence and multiplicity of solutions of singular problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ1_HTML.gif)
where is allowed to have singularities at
and
, have been studied by several authors, see Asakawa [1], Agarwal and O
Regan [2], O
Regan [3], Habets and Zanolin [4], Xu and Ma [5], Yang [6], and the references therein. The main tools in [1–6] are the method of lower and upper solutions, Leray-Schauder continuation theorem, and the fixed point index theory in cones. Recently, Ma [7] studied the existence of nodal solutions of the singular boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ2_HTML.gif)
by applying Rabinowitz's global bifurcation theorem, where is allowed to have singularities at
and
is linearizable at
as well as at
. It is the purpose of this paper to study the existence of positive solutions of (1.1), which is not necessarily linearizable.
Let be Banach space defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ3_HTML.gif)
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ4_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ5_HTML.gif)
Definition 1.1.
A function is said to be an
-Carathéodory function if it satisfies the following:
(i)for each ,
is measurable;
(ii)for a.e. ,
is continuous;
(iii)for any , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ6_HTML.gif)
In this paper, we will prove the existence of positive solutions of (1.1) by using the global bifurcation techniques under the following assumptions.
(H1) Let be an
-Carathéodory function and there exist functions
,
,
, and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ7_HTML.gif)
for some -Carathéodory functions
defined on
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ8_HTML.gif)
uniformly for a.e. , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ9_HTML.gif)
for some -Carathéodory functions
defined on
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ10_HTML.gif)
uniformly for a.e. .
(H2) for a.e.
and
.
(H3) There exists function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ11_HTML.gif)
Remark 1.2.
If ,
,
, and
, then (1.8) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ12_HTML.gif)
and (1.10) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ13_HTML.gif)
The main tool we will use is the following global bifurcation theorem for problem which is not necessarily linearizable.
Theorem A (Rabinowitz, [8]).
Let be a real reflexive Banach space. Let
be completely continuous, such that
,
. Let
, such that
is an isolated solution of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ14_HTML.gif)
for and
, where
,
are not bifurcation points of (1.14). Furthermore, assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ15_HTML.gif)
where is an isolating neighborhood of the trivial solution. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ16_HTML.gif)
then there exists a continuum (i.e., a closed connected set) of
containing
, and either
(i) is unbounded in
, or
(ii).
To state our main results, we need the following.
Lemma 1.3 (see [1, Proposition ]).
Let , then the eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ17_HTML.gif)
has a sequence of eigenvalues as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ18_HTML.gif)
Moreover, for each ,
is simple and its eigenfunction
has exactly
zeros in
.
Remark 1.4.
Note that and
for each
. Therefore, there exist constants
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ19_HTML.gif)
Our main result is the following.
Theorem 1.5.
Let (H1)–(H3) hold. Assume that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ20_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ21_HTML.gif)
then (1.1) has at least one positive solution.
Remark 1.6.
For other references related to this topic, see [9–14] and the references therein.
2. Preliminary Results
Lemma 2.1 (see [15, Proposition ]).
For any , the linear problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ22_HTML.gif)
has a unique solution and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ24_HTML.gif)
Furthermore, if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ25_HTML.gif)
Let be the Banach space with the norm
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ26_HTML.gif)
Let be an operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ27_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ28_HTML.gif)
Then, from Lemma 2.1, is well defined.
Lemma 2.2.
Let and
be the first eigenfunction of (1.17). Then for all
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ29_HTML.gif)
Proof.
For any , integrating by parts, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ30_HTML.gif)
Since and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ31_HTML.gif)
Therefore, we only need to prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ32_HTML.gif)
Let us deal with the first equality, the second one can be treated by the same way. Note that , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ33_HTML.gif)
which implies that . Then
is bounded on
. Now, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ34_HTML.gif)
Suppose on the contrary that , then for
small enough, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ35_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ36_HTML.gif)
which is a contradiction. Combining (1.19) with (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ37_HTML.gif)
This completes the proof.
Remark 2.3.
Under the conditions of Lemma 2.2, for the later convenience, (2.8) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ38_HTML.gif)
Lemma 2.4 (see [1, Lemma ]).
For every , the subset
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ39_HTML.gif)
is precompact in .
Let be the closure of the set of positive solutions of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ40_HTML.gif)
We extend the function to an
-Carathéodory function
defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ41_HTML.gif)
Then for
and a.e.
. For
, let
be an arbitrary solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ42_HTML.gif)
Since for a.e.
, Lemma 2.2 yields
for
. Thus,
is a nonnegative solution of (2.19), and the closure of the set of nontrivial solutions
of (2.21) in
is exactly
.
Let be an
-Carathéodory function. Let
be the Nemytskii operator associated with the function
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ43_HTML.gif)
Lemma 2.5.
Let on
. Let
be such that
in
,
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ44_HTML.gif)
Moreover, , whenever
.
Let be the Nemytskii operator associated with the function
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ45_HTML.gif)
Then (2.21), with , is equivalent to the operator equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ46_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ47_HTML.gif)
Lemma 2.6.
Let (H1) and (H2) hold. Then the operator is completely continuous.
Proof.
From (1.10) in (H1), there exists , such that, for a.e.
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ48_HTML.gif)
Since is an
-Carathéodory function, then there exists
, such that, for a.e.
and
,
. Therefore, for a.e.
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ49_HTML.gif)
For convenience, let . We first show that
is continuous. Suppose that
in
as
. Clearly,
as
for a.e.
and there exists
such that
for every
. It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ50_HTML.gif)
By the Lebesgue dominated convergence theorem, we have that in
as
. Thus,
is continuous.
Let be a bounded set in
. Lemma 2.4 together with (2.28) shows that
is precompact in
. Therefore,
is completely continuous.
In the following, we will apply the Leray-Schauder degree theory mainly to the mapping ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ51_HTML.gif)
For , let
, let
denote the degree of
on
with respect to
.
Lemma 2.7.
Let be a compact interval with
, then there exists a number
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ52_HTML.gif)
Proof.
Suppose to the contrary that there exist sequences and
in
in
, such that
for all
, then,
in
.
Set . Then
and
. Now, from condition (H1), we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ53_HTML.gif)
and accordingly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ54_HTML.gif)
Let and
denote the nonnegative eigenfunctions corresponding to
and
, respectively, then we have from the first inequality in (2.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ55_HTML.gif)
From Lemma 2.2, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ56_HTML.gif)
Since in
, from (1.12), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ57_HTML.gif)
By the fact that , we conclude that
in
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ58_HTML.gif)
Combining this and (2.35) and letting in (2.34), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ59_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ60_HTML.gif)
Similarly, we deduce from second inequality in (2.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ61_HTML.gif)
Thus, . This contradicts
.
Corollary 2.8.
For and
,
.
Proof.
Lemma 2.7, applied to the interval , guarantees the existence of
such that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ62_HTML.gif)
This together with Lemma 2.6 implies that for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ63_HTML.gif)
which ends the proof.
Lemma 2.9.
Suppose , then there exists
such that
with
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ64_HTML.gif)
where is the nonnegative eigenfunction corresponding to
.
Proof.
Suppose on the contrary that there exist and a sequence
with
and
in
such that
for all
. As
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ65_HTML.gif)
and in
, it concludes from Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ66_HTML.gif)
Notice that has a unique decomposition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ67_HTML.gif)
where and
. Since
on
and
, we have from (2.46) that
.
Choose such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ68_HTML.gif)
By (H1), there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ69_HTML.gif)
Therefore, for a.e. ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ70_HTML.gif)
Since , there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ71_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ72_HTML.gif)
Applying (2.51), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ73_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ74_HTML.gif)
This contradicts (2.47).
Corollary 2.10.
For and
,
.
Proof.
Let , where
is the number asserted in Lemma 2.9. As
is bounded in
, there exists
such that
,
. By Lemma 2.9, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ75_HTML.gif)
This together with Lemma 2.6 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ76_HTML.gif)
Now, using Theorem A, we may prove the following.
Proposition 2.11.
is a bifurcation interval from the trivial solution for (2.30). There exists an unbounded component
of positive solutions of (2.30) which meets
. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ77_HTML.gif)
Proof.
For fixed with
, let us take that
,
and
. It is easy to check that, for
, all of the conditions of Theorem A are satisfied. So there exists a connected component
of solutions of (2.30) containing
, and either
(i) is unbounded, or
(ii).
By Lemma 2.7, the case (ii) can not occur. Thus, is unbounded bifurcated from
in
. Furthermore, we have from Lemma 2.7 that for any closed interval
, if
, then
in
is impossible. So
must be bifurcated from
in
.
3. Proof of the Main Results
Proof of Theorem 1.5.
It is clear that any solution of (2.30) of the form yields solutions
of (1.1). We will show that
crosses the hyperplane
in
. To do this, it is enough to show that
joins
to
. Let
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ78_HTML.gif)
We note that for all
since
is the only solution of (2.30) for
and
.
Case 1.
consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ79_HTML.gif)
In this case, we show that the interval
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ80_HTML.gif)
We divide the proof into two steps.
Step 1.
We show that is bounded.
Since ,
. From (H3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ81_HTML.gif)
Let denote the nonnegative eigenfunction corresponding to
.
From (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ82_HTML.gif)
By Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ83_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ84_HTML.gif)
Step 2.
We show that joins
to
.
From (3.1) and (3.7), we have that . Notice that (2.30) is equivalent to the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ85_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ86_HTML.gif)
We divide the both sides of (3.9) by and set
. Since
is bounded in
, there exist a subsequence of
and
with
and
on
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ87_HTML.gif)
relabeling if necessary. Thus, (3.9) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ88_HTML.gif)
Let and
denote the nonnegative eigenfunctions corresponding to
and
, respectively, then it follows from the second inequality in (3.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ89_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ90_HTML.gif)
Similarly, we deduce from the first inequality in (3.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ91_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ92_HTML.gif)
So joins
to
.
Case 2.
.
In this case, if is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ93_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ94_HTML.gif)
and moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ95_HTML.gif)
Assume that is bounded, applying a similar argument to that used in Step 2 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ96_HTML.gif)
Again joins
to
and the result follows.
Remark 3.1.
Lomtatidze [13, Theorem ] proved the existence of solutions of singular two-point boundary value problems as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ97_HTML.gif)
under the following assumptions:
(A1)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ98_HTML.gif)
where satisfies the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ99_HTML.gif)
?(A2) For , let
be the solution of singular IVPs
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ100_HTML.gif)
satisfying has at least one zero in
and
has no zeros in
.
It is worth remarking that (A1)-(A2) imply Condition (1.21) in Theorem 1.5. However, Condition (1.21) is easier to be verified than (A1)-(A2) since and
are easily estimated by Rayleigh's Quotient.
The language of eigenvalue of singular linear eigenvalue problem did not occur until Asakawa [1] in 2001. The first part of Theorem 1.5 is new.
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Acknowledgments
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC 11061030, the Fundamental Research Funds for the Gansu Universities.
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Ma, R., Li, J. Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem. Bound Value Probl 2010, 458015 (2010). https://doi.org/10.1155/2010/458015
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DOI: https://doi.org/10.1155/2010/458015