2.1. Auxiliary Functions
Let assumption (1.7) hold, and let us introduce auxiliary functions
, and
as
where
,
Here, the positive constants
and
are identical with those used in boundary conditions (1.6b). Note that the function
is used in the analysis of positive and pseudo dead core solutions of problem (1.6a)-(1.6b), while the function
for its dead core solutions.
Properties of
are described in the following lemma.
Lemma 2.1.
Let assumption (1.7) hold and let
. Then
, and
is increasing on
.
Proof.
Let
be arbitrary,
. Then
, and
is increasing on
by [4, Lemma
(where
is replaced by
)]. Since
is arbitrary, the result immediately follows.
In the following lemma, we introduce functions
and
and discuss their properties.
Lemma 2.2.
Let assumption (1.7) hold. Then the following statements follow.
(i)The function
is continuous on
, and
for
.
(ii)For each
, there exists a unique
such that
and
,
for
,
.
(iii)The function
is continuous on
.
Proof.
-
(i)
Let us define
on
by
Then
. Let
and define
. Then, by (1.7),
. Hence
and consequently
, which means that
is continuous at
. Let
. We now show that
is continuous at the point
. Let us choose an arbitrary
in the interval
. Then
for
and
, where
Since
by [4, Lemma
(where 1 was replaced by
)], it follows that
is continuous at
. The continuity of
at
now follows from the fact that
is continuous at this point. Hence
is continuous on
, and from
we conclude
. Since
we have
for
.
-
(ii)
Consider the equation
, that is,
The function
is increasing on
,
, and, for
,
. Hence, for each
, there exists a unique
such that
and
. Clearly,
for
. In order to prove that
, suppose the contrary, that is, suppose that
is discontinuous at a point
,
. Then there exist sequences
in
such that
, and the sequences
are convergent,
,
,
. Let
in
and in
. This means
,
, and
by the definition of the function
, which contradicts
.
-
(iii)
By (ii),
and
for
. Hence, the function
is continuous on
and positive on
. From
we now deduce that
. Since
where
, and
on
,
, we conclude
. Hence
is continuous at
, and consequently
.
Let
be the function from Lemma 2.2(ii) defined on the interval
. From now on,
denotes the value of
at
, that is,
In the following lemma, we prove a property of
which is crucial for discussing multiple positive solutions of problem (1.6a)-(1.6b).
Lemma 2.3.
Let assumption (1.8) hold and let the function
be given by (2.5). Then there exists
such that
Proof.
Note that
d
. We deduce from [4, Lemma
(with 1 replaced by
)] that there exists an
such that
If
for some
, then (2.16) yields
Consequently, inequality (2.15) holds for such an
. If the statement of the lemma were false, then some
would exist such that
and
From the following equalities, compare (2.4),
and from
, we conclude that
Finally, from
we have
, which contradicts (2.18).
In order to discuss dead core solutions of problem (1.6a)-(1.6b) and their dead cores, we need to introduce two additional functions
and
related to
and study their properties.
Lemma 2.4.
Assume that (1.7) holds and let
be given by (2.3). Then for each
, there exists a unique
such that
The function
is continuous and decreasing on
, and the function
is continuous and increasing on
. Moreover,
.
Proof.
It follows from (1.7) that
. Also,
is increasing w.r.t. both variables,
for any
, and
,
for any
. Hence, for each
, there exists a unique
such that
. In order to prove that
is decreasing on
, assume on the contrary that
for some
. Then
which contradicts
for
. Hence,
is decreasing on
. If
was discontinuous at a point
, then there would exist sequences
and
in
such that
and
,
are convergent,
, and
with
. Taking the limits
in
and
, we obtain
,
. Consequently,
by the definition of the function
, which is not possible.
By (2.22),
and therefore,
It follows from the properties of
that the functions
,
are continuous, positive, and increasing on
. Hence (2.25) implies that
and
is increasing. Moreover,
since
d
is bounded on
.
Corollary 2.5.
Let assumption (1.7) hold. Then
and for each
satisfying the inequality
there exists a unique
such that
Proof.
The equalities
for
and
imply that
. Since the function
defined by (2.23) is continuous and increasing on
, it follows that
for
; see (2.26). Let us choose an arbitrary
satisfying (2.27). Then
. Now, the properties of
guarantee that equation
has a unique solution
. This means that (2.28) holds for a unique
.
2.2. Dependence of Solutions on the Parameter
The following two lemmas characterize the dependence of positive and dead core solutions of problem (1.6a)-(1.6b) on the parameter
.
Lemma 2.6.
Let assumption (1.7) hold and let
be a positive solution of problem (1.6a)-(1.6b) for some
. Also, let
, and
. Then
,
,
where the function
is given by (2.2).
Proof.
Since
and
for
, we conclude that
on
and
,
. By integrating the equality
over
, we obtain
and consequently, since
on
,
Finally, integrating
over
yields (2.30). Now we set
in (2.30) and obtain (2.29). Equality (2.31) follows from
and from
Remark 2.7.
Let (1.7) hold and let
be a pseudo dead core solution of problem (1.6a)-(1.6b). Then, by the definition of pseudo dead core solutions,
. We can proceed analogously to the proof of Lemma 2.6 in order to show that
where
, and
From (2.38), we finally have
. Consequently,
.
Remark 2.8.
If
, then
,
is the unique solution of problem (1.6a)-(1.6b). This solution is positive.
Lemma 2.9.
Let assumption (1.7) hold and let
be a dead core solution of problem (1.6a)-(1.6b) for some
. Moreover, let
. Then
and there exists a point
such that
for
,
where the function
is given by (2.3). Furthermore,
is the unique dead core solution of problem (1.6a)-(1.6b) with
.
Proof.
Since
is a dead core solution of problem (1.6a)-(1.6b) with
, there exists by definition, a point
such that
,
for
and
on
. Consequently,
on
, and
. We can now proceed analogously to the proof of Lemma 2.6 to show that
and (2.39) holds. Setting
in (2.39), we obtain (2.40). Also, from (1.6b),
,
equality (2.41) follows.
It remains to verify that
is the unique dead core solution of problem (1.6a)-(1.6b) with
. Let us suppose that
is another dead core solution of the above problem. Let
for
and
on
for some
. Then
for
, and consequently
on
and
. Hence, compare (2.40) and (2.41),
Since
by (2.41), and the function
is increasing and continuous on
, we deduce from (2.45) and (2.46) that
. Then (2.40) and (2.44) yield
. Therefore,
d
for
. Finally, since
for
and since by Lemma 2.1 the function
is increasing on
,
follows. This completes the proof.
2.3. Main Results
Let the function
be given by (2.5) and let us denote by
the range of the function
restricted to the interval
,
Since
by Lemma 2.2(iii),
for
and
, we can have either (i)
for
, or (ii)
for some
. For (i), we have
, while in case of (ii),
with
holds. Clearly,
.
Positive solutions of problem (1.6a)-(1.6b) are analyzed in the following theorem.
Theorem 2.10.
Let assumption (1.7) hold. Then problem (1.6a)-(1.6b) has a positive solution if and only if 
. Additionally, for each
, problem (1.6a)-(1.6b) with
has a unique positive solution
such that
and
.
Proof.
Let
be a positive solution of problem (1.6a)-(1.6b) for
. By Lemma 2.6, (2.31) holds with
and
. Furthermore, by Lemmas 2.2(ii) and 2.6,
, which together with (2.29) implies that
. Consequently, 
. For
, problem (1.6a)-(1.6b) has the unique positive solution
; see Remark 2.8. Since
, 
. Consequently, if problem (1.6a)-(1.6b) has a positive solution, then 
.
We now show that for each 
, problem (1.6a)-(1.6b) has a positive solution, and if
for some
, then problem (1.6a)-(1.6b) has a unique positive solution
such that
and
. Let us choose 
. Then
for some
. If
, then
. Consequently,
and
is the unique solution of problem (1.6a)-(1.6b). Clearly,
and
since
. Let us suppose that
. If
is a positive solution of problem (1.6a)-(1.6b) and
, then, by Lemma 2.6; see (2.30), the equality
holds for
, where
is given by (2.1). Hence, in order to prove that for
problem (1.6a)-(1.6b) has a unique positive solution
such that
and
, we have to show that the equation
has a unique solution
; this solution is a positive solution of problem (1.6a)-(1.6b), and
,
. Since
,
is increasing by Lemma 2.1, and
, (2.49) has a unique solution
. It follows from
and
that
and
. In addition,
Hence,
and
. In order to show that
is continuous at
, we set
. Then, compare (2.49),
and therefore,
Consequently,
, and so
is continuous at
, or equivalently,
. Now (2.50) indicates that
and
Moreover, by the de L'Hospital rule,
As a result
and
for
. Since
and, by (2.50),
, we have
by Lemma 2.2(ii). Thus,
satisfies (1.6b), and therefore
is a unique positive solution of problem (1.6a)-(1.6b) such that
and
.
The following theorem deals with multiple positive solutions of problem (1.6a)-(1.6b).
Theorem 2.11.
Let assumption (1.8) hold. Then
, with
given by (2.48), and for each
, there exist multiple positive solutions of problem (1.6a)-(1.6b).
Proof.
By Lemmas 2.2(iii) and 2.3,
,
, and
in a right neighbourhood of
. Hence,
. Let us choose
. Then there exist
such that
for
. Now Theorem 2.10 guarantees that problem (1.6a)-(1.6b) has positive solutions
and
such that
,
. Since
, we have
and therefore, for each
, problem (1.6a)-(1.6b) has multiple positive solutions.
Next, we present results for pseudo dead core solutions of problem (1.6a)-(1.6b). Note that here
.
Theorem 2.12.
Let assumption (1.7) hold. Then problem (1.6a)-(1.6b) has a pseudo dead core solution if and only if
Moreover, for
given by (2.56), problem (1.6a)-(1.6b) has a unique pseudo dead core solution such that
.
Proof.
Let us assume that
is a pseudo dead core solution of problem (1.6a)-(1.6b) and let
. Then, by Remark 2.7, equalities (2.36), (2.38) hold, and
. Also, (2.37) implies that
is a solution of the equation
where
and
are given by (2.1) and (2.56), respectively. The result follows by showing that equation (2.57) has a unique solution and that this solution is a pseudo dead core solution of problem (1.6a)-(1.6b). We verify these facts for solutions of (2.57) arguing as in the proof of Theorem 2.10, with
replaced by 0.
In the final theorem below, we deal with dead core solutions of problem (1.6a)-(1.6b).
Theorem 2.13.
Let assumption (1.7) hold and let
be the function defined in Lemma 2.4. Then the following statements hold.
-
(i)
Problem (1.6a)-(1.6b) has a dead core solution if and only if
-
(ii)
For each
satisfying (2.58), problem (1.6a)-(1.6b) has a unique dead core solution.
-
(iii)
If the subinterval
is the dead core of a dead core solution
of problem (1.6a)-(1.6b), then
and
Proof.
-
(i)
Let
be a dead core solution of problem (1.6a)-(1.6b) for some
and let
. Then there exists a point
such that
for
, and equalities (2.39), (2.40), and (2.41) are satisfied by Lemma 2.9. We deduce from (2.41) and from Lemma 2.4 that
. Therefore, compare (2.40),
Since
by Corollary 2.5, we have
Hence, if problem (1.6a)-(1.6b) has a dead core solution, then
satisfies inequality (2.58).
We now prove that for each
satisfying (2.58), problem (1.6a)-(1.6b) has a dead core solution. Let us choose
satisfying (2.58). Then, by Corollary 2.5, there exists a unique
such that
Let us now consider, compare (2.39),
where
is given by (2.1). Since
and
is increasing on
by Lemma 2.1,
, and, by (2.63),
, there exists a unique solution
of (2.64) and
,
. In addition,
and consequently,
and
. Since
by the Mean Value Theorem for integrals, where
, we have
Therefore,
since
. Hence,
is continuous at
, and
. Furthermore,
Let
Then
,
for
,
,
, and
Thus,
where
is given by (2.3). Since
by Lemma 2.4,
satisfies the boundary conditions (1.6b). Consequently,
is a dead core solution of problem (1.6a)-(1.6b).
-
(ii)
Let us choose an arbitrary
satisfying (2.58). By (i), problem (1.6a)-(1.6b) has a dead core solution which is unique by Lemma 2.9.
-
(iii)
Let the subinterval
be the dead core of a dead core solution
of problem (1.6a)-(1.6b). Then, by Lemma 2.9, equalities (2.40) and (2.41) hold with
replaced by
and
. Since
by the definition of the function
, we have
. Equality (2.59) now follows from (2.40) with
and
replaced by
and
, respectively.
Example 2.14.
We now turn to the case study of the boundary value problem (1.9a)-(1.9b),
Note that (1.9a)-(1.9b) is a special case of (1.6a)-(1.6b) with
satisfying (1.8). Since
we have
for
, and
for
. By Lemma 2.2, the equation
has a unique solution
for
,
,
for
, and
. Let
Then
,
, and
. In order to show that
is increasing on
it is sufficient to verify that
is injective. Let us assume that this is not the case, then there exist
,
, such that
. From
,
, or equivalently, from
it follows that
, and
which is a contradiction. Hence,
is increasing on
and therefore, there exists the inverse function
mapping
onto
. Since
and
for
, we have
Consequently,
In order to discuss the range
of the function
and the value of
for
, we first consider properties of the function
defined on
. Let
Then
where
. The function
vanishes only at point
in the interval
, and
, because
and
. Since
,
on
,
on
and
for
, we have
on
and
on
. Let us define
. Then
, and it follows from
that
on
and
on
. Consequently,
is increasing on
and decreasing on
. It follows from the equality
for
and from the properties of the functions
and
that
is increasing on
and decreasing on
. Hence,
, where
. Also,
Using properties of the function
and the results of Theorems 2.10–2.13, we can now characterize the structure of the solution
.
-
(i)
For each
, there exists only a unique dead core solution of problem (1.9a)-(1.9b).
-
(ii)
For
, there exist a unique dead core solution and a unique positive solution of problem (1.9a)-(1.9b).
-
(iii)
For each
, there exist a unique dead core solution and exactly two positive solutions of problem (1.9a)-(1.9b).
-
(iv)
For
, there exist the unique pseudo dead core solution
and a unique positive solution of problem (1.9a)-(1.9b).
-
(v)
For each
, there exist only a unique positive solution of problem (1.9a)-(1.9b).
Using Theorem 2.10, Lemma 2.6, and the properties of the function
, we can specify further properties of positive solutions of problem (1.9a)-(1.9b).
-
(i)
If
is the (unique) positive solution of problem (1.9a)-(1.9b) with
, then
, where
is the root of the equation
.
-
(ii)
If
are the (unique) positive solutions of problem (1.9a)-(1.9b) with
, then
,
, where
, are the roots of the equation
.
We are also able to give some more information on the dead core solutions of problem (1.9a)-(1.9b). Since
the function
,
, is the solution of the equation
. Let us choose an arbitrary
. By Corollary 2.5, the equation; see (2.28),
has a unique solution
. Consequently,
One can easily show that the function
is the unique dead core solution of problem (1.9a)-(1.9b). Additionally, it follows from Theorem 2.13(iii) that
since
.