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Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with ϕ-Laplacian
Advances in Difference Equations volume 2010, Article number: 262854 (2010)
Abstract
The paper discusses the existence of positive solutions, dead-core solutions, and pseudo-dead-core solutions of the singular problem ,
,
. Here
is a positive parameter,
,
,
,
,
is singular at
and
may be singular at
.
1. Introduction
Consider the singular boundary value problem


depending on the parameter . Here
,
satisfies the Carathéodory conditions on
,
(
,
is positive,
for a.e.
and each
, and
may be singular at
.
Throughout the paper denotes the set of absolutely continuous functions on
and
is the norm in
.
We investigate positive, dead-core, and pseudo-dead-core solutions of problem (1.1), (1.2).
A function is a positive solution of problem (1.1), (1.2) if
,
on
,
satisfies (1.2), and (1.1) holds for a.e.
.
We say that satisfying (1.2) is a dead-core solution of problem (1.1), (1.2) if there exist
such that
on
,
on
,
and (1.1) holds for a.e.
. The interval
is called the dead-core of
. If
, then
is called a pseudo-dead-core solution of problem (1.1), (1.2).
The existence of positive and dead core solutions of singular second-order differential equations with a parameter was discussed for Dirichlet boundary conditions in [1, 2] and for mixed and Robin boundary conditions in [3–5]. Papers [6, 7] discuss also the existence and multiplicity of positive and dead core solutions of the singular differential equation satisfying the boundary conditions
,
and
,
, respectively, and present numerical solutions. These problems are mathematical models for steady-state diffusion and reactions of several chemical species (see, e.g., [4, 5, 8, 9]). Positive and dead-core solutions to the third-order singular differential equation

satisfying the nonlocal boundary conditions ,
, were investigated in [10].
We work with the following conditions on the functions and
in the differential equation (1.1). Without loss of generality we can assume that
for each
(otherwise
is replaced by
), where
is from (1.2).
-
(H1)
is an increasing and odd homeomorphism such that
.
-
(H2)
, where
, and
(1.4) -
(H3)for a.e.
and all
,
(1.5)
where ,
,
,
, and
are positive,
are nonincreasing,
are nondecreasing,
for
, and

The aim of this paper is to discuss the existence of positive, dead-core, and pseudo-dead-core solutions of problem (1.1), (1.2). Since problem (1.1), (1.2) is singular we use regularization and sequential techniques.
For this end for , we define
, where
, and
by the formulas

Then and
give




Consider the auxiliary regular differential equation

A function is a solution of problem (1.12), (1.2) if
,
fulfils (1.2), and (1.12) holds for a.e.
.
We introduce also the notion of a sequential solution of problem (1.1), (1.2). We say that is a sequential solution of problem (1.1), (1.2) if there exists a sequence
,
, such that
in
, where
is a solution of problem (1.12), (1.2) with
replaced by
. In Section 3 (see Theorem 3.1) we show that any sequential solution of problem (1.1), (1.2) is either a positive solution or a pseudo-dead-core solution or a dead-core solution of this problem.
The next part of our paper is divided into two sections. Section 2 is devoted to the auxiliary regular problem (1.12), (1.2). We prove the solvability of this problem by the existence principle in [11] and investigate the properties of solutions. The main results are given in Section 3. We prove that under assumptions ()–(
), for each
problem (1.1), (1.2) has a sequential solution and that any sequential solution is either a positive solution or a pseudo-dead-core solution or a dead-core solution (Theorem 3.1). Theorem 3.2 shows that for sufficiently small values of
all sequential solutions of problem (1.1), (1.2) are positive solutions while, by Theorem 3.3, all sequential solutions are dead-core solutions if
is sufficiently large. An example demonstrates the application of our results.
2. Auxiliary Regular Problems
The properties of solutions of problem (1.12), (1.2) are given in the following lemma.
Lemma 2.1.
Let ()–(
) hold. Let
be a solution of problem (1.12), (1.2). Then



Proof.
Suppose that . Then
. Let

Then and, by (1.9),
a.e. on
. Hence
is increasing on
, and therefore,
is also increasing on this interval since
is increasing on
by
. Consequently,
and
on
. Then
, which contradicts
. Hence
. Let
. Then
on a right neighbourhood of
. Put

Then on
, and therefore,
a.e. on
, which implies that
is decreasing on
. Now it follows from
and
that
,
on
and
on
. Consequently,
, which contradicts
. To summarize,
and
. Suppose that
. Then there exist
such that
,
and
on
. Hence
a.e. on
and arguing as in the above part of the proof we can verify that
and
,
on
. Consequently,
, which is impossible. Hence
on
. New it follows from (1.9) and (1.10) that
a.e. on
, which together with
gives that
is nondecreasing on
. Suppose that
for some
. If
, then
, which contradicts
since
. Hence
and
. Let

Then ,
and
is increasing on
since
a.e. on this interval by (1.9). Hence there exists
,
, such that
on
and it follows from the definition of the function
that

where ,
and
a.e. on
. Integrating (2.7) over
yields

From this equality, from and from
, where
, we obtain

for . Since
for
, we have

which is impossible. We have proved that

Hence a.e. on
by (1.9), and therefore,
is increasing on
. If
, then
on
, and so
, which is impossible since
. Consequently,
and
vanishes at a unique point
. Hence (2.3) is true.
Next, we deduce from ,
and from
that
and
. Consequently,
. Hence (2.2) holds. Inequality (2.1) follows from (2.2), (2.3), and (2.11).
Remark 2.2.
Let be a solution of problem (1.12), (1.2) with
. Then
a.e. on
, and so
is a constant function. Let
. Now, it follows from (1.2) that
and
. Consequently,
, and since
, we have
. Hence
, and
is the unique solution of problem (1.12), (1.2) for
.
The following lemma gives a priori bounds for solutions of problem (1.12), (1.2).
Lemma 2.3.
Let ()–(
) hold. Then there exists a positive constant
independent of
and depending on
such that

for any solution of problem (1.12), (1.2).
Proof.
Let be a solution of problem (1.12), (1.2). By Lemma 2.1,
satisfies (2.1)–(2.3). Hence

In view of (1.11),

for a.e. and

for a.e. . Since
for
by
, we have

Therefore,

for a.e. and

for a.e. . Integrating (2.17) over
and (2.18) over
gives


respectively. We now show that condition (1.6) implies

Since by
, we have
. Therefore, there exists
such that

Then

and (2.21) follows from (1.6). Since , inequality (2.21) guarantees the existence of a positive constant
such that

for all . Hence (2.19) and (2.20) imply
. Consequently,
and equality (2.13) shows that (2.12) is true for
.
Remark 2.4.
By Lemma 2.3, estimate (2.12) is true for any solution of problem (1.12), (1.2), where
is a positive constant independent of
and depending on
. Fix
and consider the differential equation

It follows from the proof of Lemma 2.3 that for each
and any solution
of problem (2.25), (1.2). Since
is the unique solution of this problem with
by Remark 2.2, we have
for each
and any solution
of problem (2.25), (1.2).
We are now in the position to show that problem (1.12), (1.2) has a solution. Let ,
, be defined by

where and
are as in (1.2). We say that the functionals
and
are compatible if for each
the system

has a solution . We apply the following existence principle which follows from [11–13] to prove the solvability of problem (1.12), (1.2).
Proposition 2.5.
Let ()–(
) hold. Let there exist positive constants
such that

for each and any solution
of problem (2.25), (1.2). Also assume that
and
are compatible and there exist positive constants
such that

for each and each solution
of system (2.27).
Then problem (1.12), (1.2) has a solution.
Lemma 2.6.
Let ()–(
) hold. Then problem (1.12), (1.2) has a solution.
Proof.
By Lemmas 2.1 and 2.3 and Remark 2.4, there exists a positive constant such that

for each and any solution
of problem (2.25), (1.2). Hence (2.28) is true for
and
. System (2.27) has the form of

Subtracting the first equation from the second, we get . Due to
for
, we have
, and consequently,
. Hence
is the unique solution of system (2.31). Therefore,
and
are compatible and (2.29) is fulfilled for
and
. The result now follows from Proposition 2.5.
The following result deals with the sequences of solutions of problem (1.12), (1.2).
Lemma 2.7.
Let ()–(
) hold and let
be a solution of problem (1.12), (1.2). Then
is equicontinuous on
.
Proof.
By Lemmas 2.1 and 2.3, relations (2.1)–(2.3) and (2.12) hold, where is a positive constant. Let
,
and
be defined by the formulas

where and
are given in (1.11). Then
is an increasing and odd function on
,
by (2.21), and
is increasing on
. Since
is bounded in
,
is equicontinuous on
, and consequently,
is equicontinuous on
, too. Let us choose an arbitrary
. Then there exists
such that

In order to prove that is equicontinuous on
, let
and
. If
, then integrating (2.17) from
to
gives

If , then integrating (2.18) over
yields

Finally, if , then one can check that

To summarize, we have

whenever and
. Hence
is equicontinuous on
and, since
is bounded in
and
is continuous and increasing on
,
is equicontinuous on
.
The results of the following two lemmas we use in the proofs of the existence of positive and dead-core solutions to problem (1.1), (1.2).
Lemma 2.8.
Let ()–(
) hold. Then there exist
and
such that

where is any solution of problem (1.12), (1.2) with
.
Proof.
Suppose that the lemma was false. Then we could find sequences and
,
, and a solution
of the equation
satisfying (1.2) such that
, where
. Note that
on
,
on
,
and
on
for each
by Lemma 2.1. Then, by (1.11),

for a.e. ,

for a.e. , and (cf. (2.13))

Essentially, the same reasoning as in the proof of Lemma 2.3 gives that for (cf. (2.19) and (2.20))

In view of , we have
,
. Consequently,
by (2.41). We now deduce from
for
and
, and from
that
. Hence
,
, which contradicts
,
for
.
Lemma 2.9.
Let ()–(
) hold. Then for each
there exists
such that

where is any solution of problem (1.12), (1.2) with
.
Proof.
Fix and let
be as in
. Put
,

Let and choose
. If we prove that

where is any solution of problem (1.12), (1.2), then (2.43) is true since
by Lemma 2.1. In order to prove (2.45), suppose the contrary, that is suppose that there is some
such that
. The next part of the proof is broken into two cases if
or
.
Case 1.
Suppose . By Lemma 2.1,
is increasing on
. Consequently, if
, then
for
, and so

which contradicts by Lemma 2.1. Therefore,

Keeping in mind that for
, we have, by (1.8),

and therefore,

Then

which yields

Hence , which contradicts the first inequality in (2.47).
Case 2.
Suppose . Then
is positive and increasing on
by Lemma 2.1. If
, then
on
, and consequently,

which contradicts by Lemma 2.1. Hence

Since for
, the inequality in (2.48) holds a.e. on
, and therefore, the inequality in (2.49) is true for a.e.
. Integrating
over
gives

Then

Hence , which contradicts (2.53) with
.
3. Main Results and an Example
Theorem 3.1.
Suppose there are ()–(
), then the following assertions hold.
(i)For each problem (1.1), (1.2) has a sequential solution.
(ii)Any sequential solution of problem (1.1), (1.2) is either a positive solution, a pseudo-dead-core solution, or a dead-core solution.
Proof.
-
(i)
Fix
. By Lemma 2.6, for each
problem (1.12), (1.2) has a solution
. Lemmas 2.1 and 2.7 guarantee that
is bounded in
and
is equicontinuous on
. By the Arzelà-Ascoli theorem, there exist
and a subsequence
of
such that
in
. Hence
is a sequential solution of problem (1.1), (1.2).
-
(ii)
Let
be a sequential solution of problem (1.1), (1.2). Then
and
in
, where
is a solution of problem (1.12), (1.2) with
replaced by
. Hence
and
, that is,
fulfils the boundary condition (1.2). It follows from the properties of
given in Lemmas 2.1 and 2.3 that
for
,
is nondecreasing on
and
for
, where
is a positive constant. The next part of the proof is divided into two cases if
is positive, or is equal to zero.
Case 1.
Suppose that . Then there exist
and
,
such that

Hence (cf. (1.8)) for a.e.
and all
. Since
for some
by Lemma 2.1, we have
for
, and therefore,

Essentially, the same reasoning shows that

Passing if necessary to a subsequence, we may assume that is convergent, and let
. Letting
in (3.2) and (3.3) gives

Hence is the unique zero of
,
since
fulfils (1.2), and

In addition, it follows from the Fatou lemma and from the relation

that . Therefore,
. We now show that
and
fulfils (1) a.e. on
. Let us choose
. In view of (3.1), (3.4), (3.5) and Lemma 2.1, there exist
and
such that

Then (cf. (1.11))

for a.e. and
. Letting
in

yields

for by the Lebesgue dominated convergence theorem. Since
satisfying
are arbitrary and
, equality (3.10) holds for
. Essentially, the same reasoning which is now applied to
satisfying
gives

for . Hence
and
fulfills (1.1) a.e. on
. Consequently,
is a positive solution of problem (1.1), (1.2).
Case 2.
Suppose that , and let
for some
and
on
. Since
is nondecreasing on
, we have
on
,
on
and
on
. Consequently,
on
and

Furthermore, it follows from

that is integrable on the intervals
and
by the Fatou lemma. We can now proceed analogously to Case 1 with
and with
and obtain

It follows from these equalities and from on
that
and that
fulfils (1.1) a.e. on
. Hence
is a dead-core solution of problem (1.1), (1.2) if
, and
is a pseudo-dead-core solution if
.
Theorem 3.2.
Let ()–(
) hold. Then there exists
such that for each
, all sequential solutions of problem (1.1), (1.2) are positive solutions.
Proof.
Let and
be given in Lemma 2.8. Let us choose an arbitrary
. Then (2.38) holds, where
is any solution of problem (1.12), (1.2). Let
be a sequential solution of problem (1.1), (1.2). Then
in
, where
is a solution of (1.12), (1.2) with
replaced by
. Consequently,
on
by (2.38), which means that
is a positive solution of problem (1.1), (1.2) by Theorem 3.1.
Theorem 3.3.
Let ()–(
) hold. Then for each
, there exists
such that any sequential solution
of problem (1.1), (1.2) with
satisfies the equality

which means that the dead-core of contains the interval
. Consequently, all sequential solutions of problem (1.1), (1.2) are dead-core solutions for sufficiently large value of
.
Proof.
Fix . Then, by Lemma 2.9, there exists
such that

where is any solution of problem (1.12), (1.2) with
. Let us choose
and let
be a sequential solution of problem (1.1), (1.2). Then
in
, where
is a solution of problem (1.12), (1.2) with
replaced by
. It follows from (3.16) that
for
, and since
is nondecreasing on
, (3.15) holds. Consequently,
is a dead-core solution of problem (1.1), (1.2) by Theorem 3.1.
Example 3.4.
Let ,
,
,
and
be positive. Consider the differential equation

Equation (3.17) is the special case of (1.1) with and
. Since

for , where
,
fulfils
with
,
,
,
and
. Hence, by Theorem 3.1, problem (3.17), (1.2) has a sequential solution for each
, and any sequential solution is either a positive solution or a pseudo-dead-core solution or a dead-core solution. If the values of
are sufficiently small, then all sequential solutions of problem (3.17), (1.2) are positive solutions by Theorem 3.2. Theorem 3.3 guarantees that all sequential solutions of problem (3.17), (1.2) are dead-core solutions for sufficiently large values of
.
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This work was supported by the Council of Czech Government MSM 6198959214.
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Staněk, S. Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with ϕ-Laplacian. Adv Differ Equ 2010, 262854 (2010). https://doi.org/10.1155/2010/262854
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DOI: https://doi.org/10.1155/2010/262854