- Research Article
- Open Access

# Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with *ϕ*-Laplacian

- Svatoslav Staněk
^{1}Email author

**2010**:262854

https://doi.org/10.1155/2010/262854

© Svatoslav Staněk. 2010

**Received: **18 December 2009

**Accepted: **15 March 2010

**Published: **30 March 2010

## Abstract

## Keywords

## 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].

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

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.

Proof.

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.

for any solution of problem (1.12), (1.2).

Proof.

for all . Hence (2.19) and (2.20) imply . Consequently, and equality (2.13) shows that (2.12) is true for .

Remark 2.4.

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.

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.

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.

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.

where is any solution of problem (1.12), (1.2) with .

Proof.

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.

where is any solution of problem (1.12), (1.2) with .

Proof.

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.

Hence , which contradicts the first inequality in (2.47).

Case 2.

## 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.

- (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.

for . Hence and fulfills (1.1) a.e. on . Consequently, is a positive solution of problem (1.1), (1.2).

Case 2.

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.

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.

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.

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 .

## Declarations

### Acknowledgment

This work was supported by the Council of Czech Government MSM 6198959214.

## Authors’ Affiliations

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