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

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

(1.1)

(1.2)

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

(1.3)

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

(1.6)

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

(1.7)

Then and give

(1.8)

(1.9)

(1.10)

(1.11)

Consider the auxiliary regular differential equation

(1.12)

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

(2..1)

(2..2)

(2..3)

Proof.

Suppose that . Then . Let

(2..4)

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

(2..5)

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

(2..6)

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

(2..7)

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

(2..8)

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

(2..9)

for . Since for , we have

(2..10)

which is impossible. We have proved that

(2..11)

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

(2..12)

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

(2..13)

In view of (1.11),

(2..14)

for a.e. and

(2..15)

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

(2..16)

Therefore,

(2..17)

for a.e. and

(2..18)

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

(2..19)

(2..20)

respectively. We now show that condition (1.6) implies

(2..21)

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

(2..22)

Then

(2..23)

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

(2..24)

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

(2..25)

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

(2..26)

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

(2..27)

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

(2..28)

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

(2..29)

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

(2..30)

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

(2..31)

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

(2..32)

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

(2..33)

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

(2..34)

If , then integrating (2.18) over yields

(2..35)

Finally, if , then one can check that

(2..36)

To summarize, we have

(2..37)

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

(2..38)

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),

(2..39)

for a.e. ,

(2..40)

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

(2..41)

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

(2..42)

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

(2..43)

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

Proof.

Fix and let be as in . Put ,

(2..44)

Let and choose . If we prove that

(2..45)

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

(2..46)

which contradicts by Lemma 2.1. Therefore,

(2..47)

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

(2..48)

and therefore,

(2..49)

Then

(2..50)

which yields

(2..51)

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,

(2..52)

which contradicts by Lemma 2.1. Hence

(2..53)

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

(2..54)

Then

(2..55)

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

(3..1)

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

(3..2)

Essentially, the same reasoning shows that

(3..3)

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

(3..4)

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

(3..5)

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

(3..6)

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

(3..7)

Then (cf. (1.11))

(3..8)

for a.e. and . Letting in

(3..9)

yields

(3..10)

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

(3..11)

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

(3..12)

Furthermore, it follows from

(3..13)

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

(3..14)

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

(3..15)

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

(3..16)

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

(3..17)

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

(3..18)

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 .