Open Access Research Article

Superlinear Singular Problems on the Half Line

Irena Rachůnková* and Jan Tomeček

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Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, tř . 17. listopadu 12, 771 46 Olomouc, Czech Republic

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Boundary Value Problems 2010, 2010:429813  doi:10.1155/2010/429813


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2010/1/429813


Received:19 October 2010
Accepted:7 December 2010
Published:15 December 2010

© 2010 The Author(s) Irena Rachůnková and Jan Tomeček.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper studies the singular differential equation , which has a singularity at . Here the existence of strictly increasing solutions satisfying is proved under the assumption that has two zeros 0 and and a superlinear behaviour near . The problem generalizes some models arising in hydrodynamics or in the nonlinear field theory.

1. Introduction

Let us consider the problem

(11)

(12)

where is a positive real parameter.

Definition 1.1.

Let . A function satisfying (1.1) on is called a solution of ( 1.1 ) on.

Definition 1.2.

Let be a solution of (1.1) on for each . Then is called a solution of ( 1.1 ) on. If moreover fulfils conditions (1.2), it is called a solution of problem ( 1.1 ), ( 1.2 ).

Definition 1.3.

A strictly increasing solution of problem (1.1), (1.2) is called a homoclinic solution.

In this paper we are interested in the existence of strictly increasing solutions and, in particular, of homoclinic solutions. In what follows we assume

(13)

(14)

(15)

(16)

(17)

(18)

Under assumptions (1.3)–(1.8) problem (1.1), (1.2) generalizes some models arising in hydrodynamics or in the nonlinear field theory (see [15]). If a homoclinic solution exists, many important properties of corresponding models can be obtained. Note that if we extend the function in (1.1) from the half-line onto (as an even function), then any solution of (1.1), (1.2) has the same limit as and . This is a motivation for Definition 1.3. Equation (1.1) is singular at because . In [6, 7] we have proved that assumptions (1.3)–(1.8) are sufficient for the existence of strictly increasing solutions and homoclinic solutions provided

(19)

Here we assume that (1.9) is not valid. Then

(110)

and the papers [6, 8] provide existence theorems for problem (1.1), (1.2) if has a sublinear or linear behaviour near . The case that has a superlinear behaviour near is studied in this paper. To this aim we consider the initial conditions

(111)

where , and introduce the following definition.

Definition 1.4.

Let and let be a solution of (1.1) on satisfying (1.11). Then is called a solution of problem ( 1.1 ), ( 1.11 ) on. If moreover fulfils

(112)

then is called an escape solution of problem ( 1.1 ), ( 1.11 ).

We have proved in [6, 8] that for sublinear or linear the existence of a homoclinic solution follows from the existence of an escape solution of problem (1.1), (1.11). Therefore our first task here is to prove that at least one escape solution of (1.1), (1.11) exists, provided (1.3)–(1.8), (1.10), and

(113)

hold, and has a superlinear behaviour near . This is done in Section 2. Using the results of Section 2 "Theorem 2.10", and of [6, Theroms 13, 14 and 20] we get the existence of a homoclinic solution in Section 3.

Note that by Definitions 1.3 and 1.4 just the values of a solution which are less than are important for a decision whether the solution is homoclinic or escape one. Therefore condition (1.13) can be assumed without any loss of generality.

Close problems about the existence of positive solutions have been studied in [911].

2. Escape Solutions

In this section we assume that (1.3)–(1.8), (1.10), and (1.13) hold. We will need some lemmas.

Lemma 2.1 (see [6, Lemma 3]).

For each , problem (1.1), (1.11) has a unique solution on such that

(21)

In what follows by a solution of (1.1), (1.11) we mean a solution on .

Remark 2.2 (see [6, Remark 4]).

Choose and , and consider the initial conditions

(22)

Problem (1.1), (2.2) has a unique solution on . In particular, for and , we get and , respectively. Clearly, for , and are solutions of (1.1) on the whole interval .

Lemma 2.3.

Let and let be a solution of problem (1.1), (1.11) which is not an escape solution. Let us denote

(23)

Then holds and is increasing on . If , then and

(24)

Proof.

The inequality yields . By (1.1) and (1.10), we get on and hence is increasing on . As , one has on and consequently on . Therefore .

Let . Then is the first zero of and . Remark 2.2 yields that is not possible. This implies that . As is strictly increasing on and is not an escape solution, we have on . Thus on and hence is decreasing on . This gives (2.4).

Lemma 2.4.

Let and let be a solution of problem (1.1), (1.11) which is not an escape solution. Assume that is given by Lemma 2.3. Then

(25)

Proof.

From (1.1), we have

(26)

and, by multiplication and integration over ,

(27)

 (1) Assume that . The definition of yields on . Since is not an escape solution, it is bounded above and there exists

(28)

Therefore the following integral is bounded and, since it is increasing, it has a limit

(29)

So, by (2.7), exists. By virtue of (2.8), we get

(210)

If , then by (1.4), (1.10) and (2.6) we get , which contradicts (2.10). Hence, . In particular, if is defined as in Lemma 2.3, then

(211)

  (2) Assume that . Then the continuity of gives and of Lemma 2.3 fulfils . We deduce that on as in the proof of Lemma 2.3. Remark 2.2 yields that if , then neither nor can occur. Therefore .

Denote

(212)

Lemma 2.5.

Let and let be a solution of problem (1.1), (1.11). Further assume maximal such that and for . Then

(213)

For , let us denote

(214)

Then

(215)

Proof.

For equality (2.13) see Lemma 4.6 in [8]. Let us prove (2.15). Using the per partes integration, we get for

(216)

where

(217)

By multiplication and integration of (1.1) we obtain

(218)

and by the per partes integration,

(219)

To compute , we use (1.1) and get

(220)

By the per partes integration we derive

(221)

We have proved that (2.15) is valid.

Lemma 2.6.

Let , and let be solutions of problem (1.1), (1.11) with , . Let us denote

(222)

Then for each there exists a unique satisfying

(223)

If the sequence is unbounded, then there exists an escape solution in .

Proof.

Choose . The monotonicity and continuity of in give a unique . If is unbounded we argue as in the proof of Lemma 4.8 in [8].

Let and let , , and be sequences from Lemma 2.6. Assume that for any , is not an escape solution of problem (1.1), (1.11). Lemma 2.6 implies that

(224)

We can assume that that either there exists such that

(225)

or

(226)

Otherwise we take a subsequence. Some additional properties of are given in the next two lemmas.

Lemma 2.7.

Denote

(227)

and assume that the sequence is bounded above. Then there exists such that

(228)

Proof.

By Lemma 2.4 we have

(229)

Step 1 (sequence is bounded).

Assume on the contrary that is unbounded. We may write

(230)

(otherwise we take a subsequence). Equality (2.13) yields for and ,

(231)

Using (1.4), (1.6), (1.10), and the fact that for , we get

(232)

Consequently, inequality in (2.31) leads to

(233)

for . Therefore

(234)

We will consider two cases.

Case 1.

If (2.25) holds, then (2.34) gives for

(235)

By (2.30), for each sufficiently large , we get

(236)

Putting it to (2.35), we have , contrary to (2.29).

Case 2.

If (2.26) holds, then (2.34) gives for

(237)

Due to (2.30), we have

(238)

for each sufficiently large . Putting it to (2.37), we get for . Integrating it over , we obtain . Equation (1.1) and condition (1.13) yield for , and so , contrary to (2.29).

We have proved that there exists such that

(239)

Step 2 (estimate for ).

Choose . By (2.32) we get

(240)

This together with (2.31) and (2.39) imply

(241)

According to (2.27) and Lemma 2.3 we see that is the first zero of . Since the sequence is bounded above, there exists such that , . Then (1.8) and (2.41) give

(242)

Put

(243)

Then, by virtue of (2.4), inequality (2.28) is valid.

Lemma 2.8.

Consider and satisfying (2.23) and (2.24). Let , be given by (2.27). Assume that

(244)

Then there exists such that

(245)

Proof.

Assume on the contrary that

(246)

By Lemma 2.3, is increasing on , . Therefore

(247)

and therefore there exists such that

(248)

Moreover (2.23), (2.24), (2.27), (2.44), and the monotonicity of and yield

(249)

Integrating the last inequality over , we obtain , so , a contradiction.

Lemma 2.9.

Let real sequences , , be given and assume that

(250)

Let and

(251)

(for we assume ) be such that

(252)

Assume that is given by (2.14) with . Then

(253)

Proof.

By (2.50), . Condition (2.52) yields that there exists such that

(254)

Therefore

(255)

Hence

(256)

where . Consequently,

(257)

where , because is less than the critical value . We have proved (2.53).

Now we are ready to prove the following main result of this paper.

Theorem 2.10.

Assume that

(258)

for some . Further, let and be such that (2.51) and (2.52) are valid. Then there exists such that the corresponding solution of problem (1.1), (1.11) is an escape solution.

Proof.

Assumption (2.51) implies , and hence we can choose and define by (2.14). According to (1.4), (1.10), and (2.56), there exists such that for . Consequently, we can find such that

(259)

Let , , , be sequences defined in Lemma 2.6. Moreover, let

(260)

Assume that for any , is not an escape solution of problem (1.1), (1.11). By Lemma 2.4 we have

(261)

Condition (2.60) gives such that

(262)

Choose an arbitrary . We will construct a contradiction.

Step 1 (inequality for ).

Since is increasing on , (2.62) gives a unique satisfying

(263)

By (2.59) we have

(264)

because for . Further, there exists satisfying

(265)

Therefore, according to (2.12),

(266)

Let us put

(267)

Then inequalities (2.15) and (2.66) imply

(268)

Step 2 (estimate of from below).

Since is a solution of (1.1) on , we have

(269)

Therefore

(270)

where , are such that

(271)

Integrating (2.70) over , we get

(272)

Hence

(273)

By (2.52), (2.60) and , we deduce that

(274)

Since for , we get

(275)

Due to (2.58), there exists such that . Then and . Hence for each there exists such that, for ,

(276)

Consequently

(277)

Having in mind (2.75), we can choose in (2.62) such that for all the inequality holds. Hence (2.72) and the first inequality in (2.77) yield

(278)

Put . Then , and

(279)

On the other hand, by (2.76),

(280)

By (2.79), this yields

(281)

Step 3 (estimate of ).

The inequality gives . Hence there exists such that

(282)

Having in mind (2.76), we choose to this and then, by the second inequality in (2.77), we obtain

(283)

Therefore

(284)

By Lemmas 2.7 and 2.8 there exists such that

(285)

Here , , if (2.25) holds and , , if (2.26) holds. In addition there exists such that , . (Note that if in Lemma 2.8 is not bounded but does not fulfil (2.44), we work with a proper subsequence fulfilling (2.44).) By virtue of (2.84) and (2.85) we get

(286)

Inequalities (2.84) and (2.86) yield

(287)

Step 4 (final contradictions).

Putting (2.81) and (2.87) to (2.68) and using (1.6), (1.10) and , we obtain

(288)

First, let us assume that (2.26) holds and , . So, conditions (2.85), and (2.88) yield

(289)

Letting we get a contradiction to (2.53).

Finally, let us assume that (2.25) holds and , . Then (2.61), (2.88), and yield

(290)

contrary to (2.53).

Remark 2.11.

We assume that in Theorem 2.10. In particular for and , , the function can behave in neighbourhood of as a function for arbitrary .

Now, let (2.58) hold for . Then and therefore

(291)

which is the first condition in (1.9). We have proved in [6, 7] that, in this case, assumptions (1.3)–(1.8) are sufficient for the existence of an escape solution.

Example 2.12.

Let and , . Then and

(292)

Hence, for condition (2.58) is satisfied. The critical value is equal to 3. By Theorem 2.10, if fulfils (2.52) with , problem (1.1), (1.11) has an escape solution.

Example 2.13.

Let , . Then and

(293)

Hence, for condition (2.58) is satisfied. The critical value is equal to 5. By Theorem 2.10, if fulfils (2.52) with , problem (1.1), (1.11) has an escape solution.

3. Homoclinic Solutions

Having an escape solution we can deduce the existence of a homoclinic solution by the same arguments as in [6]. For completeness we bring here the main ideas. Remember that our basic assumptions (1.3)–(1.8), (1.10) and (1.13) are fulfilled in this section.

By Lemma 11 in [6], a solution of problem (1.1), (1.11) is homoclinic if and only if

(31)

By Theorem 16 in [6], a solution of problem (1.1), (1.11) is an escape solution if and only if

(32)

The third type of solutions of problem (1.1), (1.11) is characterized in the next definition.

Definition 3.1.

A solution of problem (1.1), (1.11) is called damped, if

(33)

The following properties of damped and escape solutions are important for the existence of homoclinic solutions.

Theorem 3.2 (see [6, Theorem 13] (on damped solutions)).

Let be of (1.5) and (1.6). Assume that is a solution of problem (1.1), (1.11) with . Then is damped.

Theorem 3.3 (see [6, Theorem 14]).

Let be the set of all such that corresponding solutions of problem (1.1), (1.11) are damped. Then is open in .

Theorem 3.4 (see [6, Theorem 20]).

Let be the set of all such that corresponding solutions of problem (1.1), (1.11) are escape ones. Then is open in .

Having these theorems we get the main result of this section.

Theorem 3.5 (On a homoclinic solution).

Assume that the assumptions of Theorem 2.10 are satisfied. Then problem (1.1), (1.2) has a homoclinic solution.

Proof.

By Theorems 3.2 and 3.3, the set is nonempty and open in . By Theorem 3.4, the set is open in . Using Theorem 2.10, we get that is nonempty. Therefore the set is nonempty and if , then the corresponding solution of problem (1.1), (1.11) is neither damped nor an escape solution. Therefore , and by Lemma 11 in [6], such solution is homoclinic.

The proof of Theorem 3.5 implies that if problem (1.1), (1.11) has an escape solution, then it has also a homoclinic solution. Hence the following corollary is true.

Corollary 3.6.

Assume that the assumptions of Theorem 2.10 are satisfied. Let problem (1.1), (1.11) have no homoclinic solution. Then it has no escape solution.

If we assume (2.51) and (2.52), then the growth of at is less than the critical value . This is necessary for the existence of homoclinic solutions of some types of (1.1). See the next example.

Example 3.7.

Let , , . Consider (1.1), where and for and for . Then and satisfy conditions (1.3)–(1.8), (1.10), (1.13), (2.52) and (2.58) with . By Theorem 3.5, if

(34)

then problem (1.1), (1.11) has a homoclinic solution. But if

(35)

then we have proved in [12] that problem (1.1), (1.11) has no homoclinic solution and consequently no escape solution.

Acknowledgment

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

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