Abstract
In this paper, we investigate the singular Sturm-Liouville problem 
, 
, 
, where 
is a nonnegative parameter, 
, 
, and 
. We discuss the existence of multiple positive solutions and show that for certain values of 
, there also exist solutions that vanish on a subinterval 
, the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for 
and for some model problems from the class of singular differential equations 
discussed in Agarwal et al. (2007). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.
. Here 
is the concentration of one of the reactants and 
is the Thiele modulus. In case that 
is radial symmetric with respect to 
, the radial solutions of the above equation satisfying the boundary conditions
denotes the radial coordinate. Baxley and Gersdorff [
and 
were continuous and 
was allowed to be unbounded for 
. They proved the existence of positive solutions and dead core solutions (vanishing on a subinterval 
, 
) of problem (1.3), and also covered the case of the function 
approximated by some regular function 
.
-Laplacian,
is a parameter, the function 
is non-negative and satisfies the Carathéodory conditions on 
, 
for a.e. 
, and 
is positive and satisfies the Carathéodory conditions on 
, 
. Moreover, the function 
is singular at 
and 
is singular at 
.
the set of functions 
which are absolutely continuous on 
for arbitrary small 
.
is called a positive solution of problem (1.4a)-(1.4b) if 
on 
, 
, 
satisfies (1.4b) and (1.4a) holds for a.e. 
. We say that 
satisfying (1.4b) is a dead core solution of problem (1.4a)-(1.4b) if there exists a point 
such that 
on 
, 
on 
, 
and (1.4a) holds for a.e. 
. The interval 
is called the dead core of
. If 
, 
on 
, 
, 
satisfies (1.4b) and (1.4a) holds a.e. on 
, then 
is called a pseudo dead core solution of problem (1.4a)-(1.4b).
, and 
were specified which guarantee that for each 
, problem (1.4a)-(1.4b) has a sequential solution and that any sequential solution is either a positive solution, a pseudo dead core solution, or a dead core solution. Also, it was shown that all sequential solutions of (1.4a)-(1.4b) are positive solutions for sufficiently small positive values of 
and dead core solutions for sufficiently large values of 
.
, and 
. We note that in papers [
is a non-negative parameter, and the function 
becomes unbounded at 
. Problem (1.6a)-(1.6b) is the special case of problem (1.4a)-(1.4b).
is a positive solution of problem (1.6a)-(1.6b) if 
satisfies the boundary conditions (1.6b), 
on 
and (1.6a) holds for 
. A function 
is called a dead core solution of problem (1.6a)-(1.6b) if there exists a point 
such that 
for 
, 
, 
satisfies (1.6b) and (1.6a) holds for 
. The interval 
is called the dead core of
. If 
, then 
is called a pseudo dead core solution of problem (1.6a)-(1.6b).
and the number and types of solutions to problem (1.6a)-(1.6b), provided that
to the singular boundary value problem 
satisfying (1.8).
, 
have been discussed. For further results on existence of positive and dead core solutions to differential equations of the types 
and 
, we refer the reader to [
, 
, 
, 
.
satisfies (1.7). By Lemma 2.2, the equation 
has a unique continuous solution 
, and the function
. Let 
. Then the following statements hold.
. In addition, for each 
, problem (1.6a)-(1.6b) with 
has a unique positive solution such that 
, 
.
, problem (1.6a)-(1.6b) has a unique dead core solution.
. Then 
and for each 
, there exist multiple positive solutions of problem (1.6a)-(1.6b).
. The numerical treatment of problems (1.9a)-(1.9b) and (1.5a)-(1.5b) based on the collocation method is discussed in Section 3, where for different values of 
, we study positive, pseudo dead core, and dead core solutions of problem (1.9a)-(1.9b) and positive solutions of problem (1.5a)-(1.5b).
, and 
as
,
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.
are described in the following lemma.
. Then 
, and 
is increasing on 
.
be arbitrary, 
. Then 
, and 
is increasing on 
by [
(where 
is replaced by 
)]. Since 
is arbitrary, the result immediately follows.
and 
and discuss their properties.
is continuous on 
, and 
for 
.
, there exists a unique 
such that
, 
for 
, 
.
.
on 
by 
. Let 
and define 
. Then, by (1.7), 
. Hence
, 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
by [
(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
for 
.
, that is, 
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 
.
and 
for 
. Hence, the function 
is continuous on 
and positive on 
. From
. Since
, and 
on 
, 
, we conclude 
. Hence 
is continuous at 
, and consequently 
.
be the function from Lemma 2.2(ii) defined on the interval 
. From now on, 
denotes the value of 
at 
, that is,
which is crucial for discussing multiple positive solutions of problem (1.6a)-(1.6b).
be given by (2.5). Then there exists 
such that
d
. We deduce from [
(with 1 replaced by 
)] that there exists an 
such that
for some 
, then (2.16) yields
. If the statement of the lemma were false, then some 
would exist such that 
and
, we conclude that
, which contradicts (2.18).
and 
related to 
and study their properties.
be given by (2.3). Then for each 
, there exists a unique 
such that
is continuous and decreasing on 
, and the function
. Moreover, 
.
. 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.
that the functions 
, 
are continuous, positive, and increasing on 
. Hence (2.25) implies that 
and 
is increasing. Moreover, 
since 
d
is bounded on 
.
satisfying the inequality
such that
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 
.
.
be a positive solution of problem (1.6a)-(1.6b) for some 
. Also, let 
, and 
. Then 
, 
,
is given by (2.2).
and 
for 
, we conclude that 
on 
and 
, 
. By integrating the equality 
over 
, we obtain
on 
,
yields (2.30). Now we set 
in (2.30) and obtain (2.29). Equality (2.31) follows from 
and from
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
, and
. Consequently, 
.
, then 
, 
is the unique solution of problem (1.6a)-(1.6b). This solution is positive.
be a dead core solution of problem (1.6a)-(1.6b) for some 
. Moreover, let 
. Then 
and there exists a point 
such that 
for 
,
is given by (2.3). Furthermore, 
is the unique dead core solution of problem (1.6a)-(1.6b) with 
.
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
in (2.39), we obtain (2.40). Also, from (1.6b), 
,
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),
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.
be given by (2.5) and let us denote by 
the range of the function 
restricted to the interval 
,
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
.
. Additionally, for each 
, problem (1.6a)-(1.6b) with 
has a unique positive solution 
such that 
and 
.
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 
.
, 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
; 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,
and 
. In order to show that 
is continuous at 
, we set 
. Then, compare (2.49),
, and so 
is continuous at 
, or equivalently, 
. Now (2.50) indicates that 
and
and 
for 
. Since 
and, by (2.50), 
, we have
satisfies (1.6b), and therefore 
is a unique positive solution of problem (1.6a)-(1.6b) such that 
and 
.
, with 
given by (2.48), and for each 
, there exist multiple positive solutions of problem (1.6a)-(1.6b).
, 
, 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.
.
given by (2.56), problem (1.6a)-(1.6b) has a unique pseudo dead core solution such 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
and 
are given by (2.1) and (2.56), respectively. The result follows by showing that equation (
replaced by 0.
be the function defined in Lemma 2.4. Then the following statements hold.
satisfying (2.58), problem (1.6a)-(1.6b) has a unique dead core solution.
is the dead core of a dead core solution 
of problem (1.6a)-(1.6b), then 
and
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), 
satisfies inequality (2.58).
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
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 
. Since
, we have
. Hence, 
is continuous at 
, and 
. Furthermore,
, 
for 
, 
, 
, and
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).
satisfying (2.58). By (i), problem (1.6a)-(1.6b) has a dead core solution which is unique by Lemma 2.9.
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.
satisfying (1.8). Since
, and 
for 
. By Lemma 2.2, the equation 
has a unique solution 
for 
, 
, 
for 
, and 
. Let
, 
, 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
, and 
which is a contradiction. Hence, 
is increasing on 
and therefore, there exists the inverse function 
mapping 
onto 
. Since
for 
, we have
of the function 
and the value of 
for 
, we first consider properties of the function
. Let
. The function 
vanishes only at point
, and 
, because 
and 
. Since 
, 
on 
, 
on 
and
, 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,
and the results of Theorems 2.10–2.13, we can now characterize the structure of the solution 
.
, there exists only a unique dead core solution of problem (1.9a)-(1.9b).
, there exist a unique dead core solution and a unique positive solution of problem (1.9a)-(1.9b).
, there exist a unique dead core solution and exactly two positive solutions of problem (1.9a)-(1.9b).
, there exist the unique pseudo dead core solution 
and a unique positive solution of problem (1.9a)-(1.9b).
, there exist only a unique positive solution of problem (1.9a)-(1.9b).
, we can specify further properties of positive solutions of problem (1.9a)-(1.9b).
is the (unique) positive solution of problem (1.9a)-(1.9b) with 
, then 
, where 
is the root of the equation 
.
are the (unique) positive solutions of problem (1.9a)-(1.9b) with 
, then 
, 
, where 
, are the roots of the equation 
.
, 
, is the solution of the equation 
. Let us choose an arbitrary 
. By Corollary 2.5, the equation; see (2.28),
. Consequently,
since 
.
is a collocation function
in case that the order of the underlying differential equation is 
. Here, 
are polynomials of maximal degree 
which satisfy the system (3.2a) at 
inner collocation points
, where 
is the maximal stepsize, 
To increase efficiency, an adaptive mesh selection strategy based on an a posteriori estimate for the global error of the collocation solution is utilized. A more detailed description of the numerical approach can be found in [
is unknown:
is given. The path following strategy can also cope with turning points in the path. The theoretical justification for the path following strategy implemented in 
, 
, 
, the problem has one or more positive solutions, a pseudo dead core solution or a dead core solution. All numerical approximations have been calculated on a fixed mesh with 
subintervals and collocation degree 
. Figure 
for our choice of parameters used in the following sections. Here, 
is given by (2.81).
for 
(a) and for 
, 
(b).
), 
, there exist a unique positive solution. This solution was found numerically by using the original problem formulation (1.9a)-(1.9b). For 
we obtain 
. In Figure 
. The residual 
is calculated by substituting the numerical solution 
into the differential equation,
, 
and 
.
where 
is a root of 
. Here, we have 
and 
which again shows the high quality of the numerical solution. In Figure 
, 
and 
. For this choice of parameters 
and 
. 
, 
and 
.
there exists a unique positive solution. To compute its numerical approximation, we rewrite the problem (1.9a)-(1.9b) and consider
, 
, and 
, 
are shown in Figure 
, 
and 
.
, 
and 
.
, so 
. Moreover, for the second set of parameters, 
and 
.
with 
there exist two positive solutions. These two different solutions for a fixed value of 
can be characterized via the roots 
of 
for 
. The choice of parameters remains the same. For 
, 
and 
the solution corresponding to 
is shown in Figure 
is depicted in Figure 
and 
we have 
.
, 
and 
. The associated root is 
.
, 
and 
. The associated root is 
.
as the right-hand side. For the second solution it was necessary to rewrite the problem again and use
as a free unknown parameter and 
as a necessary additional boundary condition. Here, 
and 
. For comparison, 
and 
. In Figures 
, 
, and 
, are shown. Note that 
, 
and 
. For this example 
and 
. Here, 
and 
. Finally, for 
, there exists a unique positive solution. In Figures 
, 
and for 
, 
, respectively. In this example, 
and 
. Using this latter set of parameters, we obtain 
and 
. All positive solutions could be easily found and they all show a very satisfactory level of accuracy.
, 
and 
. The associated root is 
.
, 
and 
. The associated root is 
.
, 
and 
.
, 
and 
.
. Otherwise, the problem formulated as (3.1) or (3.8), would have not been well defined at all points 
such that 
. In Figures 
, 
and 
.
, 
and 
.
where 
is the numerical solution at 
.
and 
, where 
is such that the solution vanishes on 
, is given by
and 
, in order to solve the problem,
which is not available in general. Therefore, it is especially important to note that we were able to find the dead core solution without explicit knowledge of 
by treating the problem (3.10), formulated on the whole interval 
,
, 
, and two values of 
, 
and 
, respectively. In Figures 
, 
, and 
, 
, respectively, can be found.
, 
and 
.
, 
and 
.
, 
and 
.
, 
and 
.
for a wide range of parameters. Obviously, dead core solutions can be found without exact use of the known solution structure, but the initial profile must be chosen carefully to guarantee the Newton iteration to convergence.
, 
, 
and 
, 
, we were able to solve the problem and provide the error estimate and the residual for its approximative solution. We have applied the path following strategy implemented in 
for which we were able to calculate the associated numerical solutions, are shown in Figure 
In a certain region below this value, there exist for any 
two different positive solutions.
path obtained in 76 steps of the path following procedure, where 
. The turning point has been found at 
.
.
and used 
as an initial profile. For each further step, we used the solution from the previous step as an initial profile. The solution corresponding to the values of 
shown in Figures 
we have found two different positive solutions, compare Figures 
, two different positive solutions exist; see Figures 
do not; see Figures 
.
.
.
.
.
.
.
.
.
.
-Laplacian. Boundary Value Problems 2007, 2007:-16.
-Laplacian. Computers & Mathematics with Applications 2007,54(2):255-266. 10.1016/j.camwa.2006.12.026
-Laplacian. Applications of Mathematics 2008,53(4):381-399. 10.1007/s10492-008-0031-z