Superlinear Singular Problems on the Half Line
© Rachunková and J. Tomecek. 2010
Received: 19 October 2010
Accepted: 7 December 2010
Published: 15 December 2010
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.
A strictly increasing solution of problem (1.1), (1.2) is called a homoclinic solution.
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.
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]).
Remark 2.2 (see [6, Remark 4]).
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).
We have proved that (2.15) is valid.
Choose . The monotonicity and continuity of in give a unique . If is unbounded we argue as in the proof of Lemma 4.8 in .
We will consider two cases.
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).
Then, by virtue of (2.4), inequality (2.28) is valid.
Now we are ready to prove the following main result of this paper.
Step 4 (final contradictions).
contrary to (2.53).
3. Homoclinic Solutions
Having an escape solution we can deduce the existence of a homoclinic solution by the same arguments as in . 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.
The third type of solutions of problem (1.1), (1.11) is characterized in the next definition.
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)).
Theorem 3.3 (see [6, Theorem 14]).
Theorem 3.4 (see [6, Theorem 20]).
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.
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 , 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.
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.
then we have proved in  that problem (1.1), (1.11) has no homoclinic solution and consequently no escape solution.
This paper was supported by the Council of Czech Government MSM 6198959214.
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