1. Introduction

For , , and , consider the equation where Let us put Moreover, we assume that fulfils and denote Due to (1.2)–(1.4), we see that is decreasing and positive on and increasing and positive on .

Equation (1.1) arises in many areas. For example, in the study of phase transitions of Van der Waals fluids [1–3], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [4, 5], in the homogenous nucleation theory [6], and in relativistic cosmology for description of particles which can be treated as domains in the universe [7], in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. Numerical simulations of solutions of (1.1), where is a polynomial with three zeros, have been presented in [9–11]. Close problems about the existence of positive solutions can be found in [12–14].

In this paper, we investigate a generalization of (1.1) of the form where satisfies (1.2)–(1.5) and fulfils Equation (1.7) is singular in the sense that . If , with , then satisfies (1.8), (1.9), and (1.7) is equal to (1.1).

Definition 1.1. A function which satisfies (1.7) for all is called a solution of (1.7).

Consider a solution of (1.7). Since , we have and the assumption, yields . We can find and such that for . Integrating (1.7), we get Consequently, the condition is necessary for each solution of (1.7). Denote

Definition 1.2. Let be a solution of (1.7). If , then is called a damped solution.

If a solution of (1.7) satisfies or , then we call a bounding homoclinic solution or an escape solution. These three types of solutions have been investigated in [15–18]. Here, we continue the investigation of the existence and asymptotic properties of damped solutions. Due to (1.11) and Definition 1.2, it is reasonable to study solutions of (1.7) satisfying the initial conditions Note that if , then a solution of the problem (1.7), (1.13) satisfies , and consequently is not a damped solution. Assume that , then , and if we put , a solution of (1.7), (1.13) is a constant function equal to on . Since we impose no sign assumption on for , we do not consider the case . In fact, the choice of between two zeros and 0 of has been motivated by some hydrodynamical model in [11].

A lot of papers are devoted to oscillatory solutions of nonlinear differential equations. Wong [19] published an account on a nonlinear oscillation problem originated from earlier works of Atkinson and Nehari. Wong's paper is concerned with the study of oscillatory behaviour of second-order Emden-Fowler equations where is nonnegative and absolutely continuous on . Both superlinear case () and sublinear case () are discussed, and conditions for the function giving oscillatory or nonoscillatory solutions of (1.14) are presented; see also [20]. Further extensions of these results have been proved for more general differential equations. For example, Wong and Agarwal [21] or Li [22] worked with the equation where is a positive quotient of odd integers, is positive, , , , for all . Kulenović and Ljubović [23] investigated an equation where , , or for all . The investigation of oscillatory and nonoscillatory solutions has been also realized in the class of quasilinear equations. We refer to the paper [24] by Ho, dealing with the equation where , , , , , .

Oscillation results for the equation where are positive, can be found in [25]. We can see that the nonlinearity in (1.14) is an increasing function on having a unique zero at .

Nonlinearities in all the other (1.15)–(1.18) have similar globally monotonous behaviour. We want to emphasize that, in contrast to the above papers, the nonlinearity in our (1.7) needs not be globally monotonous. Moreover, we deal with solutions of (1.7) starting at a singular point , and we provide an interval for starting values giving oscillatory solutions (see Theorems 2.3, 2.10, and 2.16). We specify a behaviour of oscillatory solutions in more details (decreasing amplitudes—see Theorems 2.10 and 2.16), and we show conditions which guarantee that oscillatory solutions converge to 0 (Theorem 3.1).

The paper is organized in this manner: Section 2 contains results about existence, uniqueness, and other basic properties of solutions of the problem (1.7), (1.13). These results which mainly concern damped solutions are taken from [18] and extended or modified a little. We also provide here new conditions for the existence of oscillatory solutions in Theorem 2.16. Section 3 is devoted to asymptotic properties of oscillatory solutions, and the main result is contained in Theorem 3.1.

2. Solutions of the Initial Problem (1.7), (1.13)

Let us give an account of this section in more details. The main objective of this paper is to characterize asymptotic properties of oscillatory solutions of the problem (1.7), (1.13). In order to present more complete results about the solutions, we start this section with the unique solvability of the problem (1.7), (1.13) on (Theorem 2.1). Having such global solutions, we have proved (see papers [15–18]) that oscillatory solutions of the problem (1.7), (1.13) can be found just in the class of damped solutions of this problem. Therefore, we give here one result about the existence of damped solutions (Theorem 2.3). Example 2.5 shows that there are damped solutions which are not oscillatory. Consequently, we bring results about the existence of oscillatory solutions in the class of damped solutions. This can be found in Theorem 2.10, which is an extension of Theorem 3.4 of [18] and in Theorem 2.16, which are new. Theorems 2.10 and 2.16 cover different classes of equations which is illustrated by examples.

Theorem 2.1 (existence and uniqueness). Assume that (1.2)–(1.5), (1.8), (1.9) hold and that there exists such that then the initial problem (1.7), (1.13) has a unique solution . The solution satisfies

Proof. Let , then the assertion is contained in Theorem  2.1 of [18]. Now, assume that , then the proof of Theorem  2.1 in [18] can be slightly modified.

For close existence results, see also Chapters 13 and 14 of [26], where this kind of equations is studied.

Remark 2.2. Clearly, for and , the problem (1.7), (1.13) has a unique solution and , respectively. Since , no solution of the problem (1.7), (1.13) with or can touch the constant solutions and .
In particular, assume that , , is a solution of the problem (1.7), (1.13) with , , and (1.2), (1.8), and (1.9) hold. If , then , and if , then .

The next theorem provides an extension of Theorem 2.4 in [18].

Theorem 2.3 (existence of damped solutions). Assume that (1.2)–(1.5), (1.8), and (1.9) hold, then for each , the problem (1.7), (1.13) has a unique solution. This solution is damped.

Proof. First, assume that there exists such that satisfies (2.1), then, by Theorem 2.1, the problem (1.7), (1.13) has a unique solution satisfying (2.2). Assume that is not damped, that is, By (1.3)–(1.5), the inequality holds. Since fulfils (1.7), we have Multiplying (2.4) by and integrating between 0 and , we get and consequently By (2.3), we can find that such that , (), and hence, according to (1.5), which is a contradiction. We have proved that , that is, is damped. Consequently, assumption (2.1) can be omitted.

Example 2.4. Consider the equation which is relevant to applications in [9–11]. Here, , , , and . Hence for , for , and Consequently, is decreasing and positive on and increasing and positive on . Since and , there exists a unique such that . We can see that all assumptions of Theorem 2.3 are fulfilled and so, for each , the problem (2.8), (1.13) has a unique solution which is damped. We will show later (see Example 2.11), that each damped solution of the problem (2.8), (1.13) is oscillatory.

In the next example, we will show that damped solutions can be nonzero and monotonous on with a limit equal to zero at . Clearly, such solutions are not oscillatory.

Example 2.5. Consider the equation where We see that in (2.10) and the functions and satisfy conditions (1.2)–(1.5), (1.8), and (1.9) with . Clearly, . Further, Since , assumption (1.5) yields and . By Theorem 2.3, for each , the problem (2.10), (1.13) has a unique solution which is damped. On the other hand, we can check by a direct computation that for each the function is a solution of equation (2.10) and satifies conditions (1.13). If , then on , and if , then , on . In both cases, .

In Example 2.5, we also demonstrate that there are equations fulfilling Theorem 2.3 for which all solutions with , not only those with , are damped. Some additional conditions giving, moreover, bounding homoclinic solutions and escape solutions are presented in [15–17].

In our further investigation of asymptotic properties of damped solutions the following lemmas are useful.

Lemma 2.6. Assume (1.2), (1.8), and (1.9). Let be a damped solution of the problem (1.7), (1.13) with which is eventually positive or eventually negative, then

Proof. Let be eventually positive, that is, there exists such that Denote .
Let , then and, by Remark 2.2, . Assume that on , then is increasing on , and there exists . Multiplying (2.4) by , integrating between and , and using notation (1.4), we obtain Letting , we get Since the function is positive and increasing, it follows that it has a limit at , and hence there exists also . If , then , which is a contradiction. Consequently Letting in (2.4) and using (1.2), (1.9) and , we get , and so , which is contrary to (2.18). This contradiction implies that the inequality on cannot be satisfied and that there exists such that . Since on , we get by (1.2), (1.7), and (1.13) that on . Due to , we see that on . Therefore, is decreasing on and . Using (2.16) with in place of , we deduce as above that (2.18) holds and that . Consequently, . We have proved that (2.14) holds provided .
If , then we take and use the above arguments. If is eventually negative, we argue similarly.

Lemma 2.7. Assume (1.2)–(1.5), (1.8), (1.9), and Let be a solution of the problem (1.7), (1.13) with , then there exists such that

Proof. Assume that such does not exist, then is positive on and, by Lemma 2.6, satisfies (2.14). We define a function By (2.19), we have and By (1.9) and (2.19), we get Since is positive on , conditions (2.14) and (2.20) yield Consequently, there exist and such that By (2.22), is positive on and, due to (2.24) and (2.27), we get Thus, is decreasing on and . If , then , contrary to the positivity of . If , then on and for . Then (2.28) yields for . We get which contradicts . The obtained contradictions imply that has at least one zero in . Let be the first zero of . Then on and, by (1.2) and (1.7), on . Due to Remark 2.2, we have also .

For negative starting value, we can prove a dual lemma by similar arguments.

Lemma 2.8. Assume (1.2)–(1.5), (1.8), (1.9), (2.19) and Let be a solution of the problem (1.7), (1.13) with , then there exists such that

The arguments of the proof of Lemma 2.8 can be also found in the proof of Lemma 3.1 in [18], where both (2.20) and (2.29) were assumed. If one argues as in the proofs of Lemmas 2.7 and 2.8 working with , and , in place of 0, and , one gets the next corollary.

Corollary 2.9. Assume (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29). Let be a solution of the problem (1.7), (1.13) with .
(I) Assume that there exist and such that then there exists such that
(II) Assume that there exist and such that then there exists such that

Note that if all conditions of Lemmas 2.7 and 2.8 are satisfied, then each solution of the problem (1.7), (1.13) with has at least one simple zero in . Corollary 2.9 makes possible to construct an unbounded sequence of all zeros of any damped solution . In addition, these zeros are simple (see the proof of Theorem 2.10). In such a case, has either a positive maximum or a negative minimum between each two neighbouring zeros. If we denote sequences of these maxima and minima by and , respectively, then we call the numbers amplitudes of .

In [18], we give conditions implying that each damped solution of the problem (1.7), (1.13) with has an unbounded set of zeros and decreasing sequence of amplitudes. Here, there is an extension of this result for .

Theorem 2.10 (existence of oscillatory solutions I). Assume that (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29) hold, Then each damped solution of the problem (1.7), (1.13) with is oscillatory and its amplitudes are decreasing.

Proof. For , the assertion is contained in Theorem  3.4 of [18]. Let be a damped solution of the problem (1.7), (1.13) with . By (2.2) and Definition 1.2, we can find such that Step 1. Lemma 2.7 yields satisfying (2.21). Hence, there exists a maximal interval such that on . If , then is eventually negative and decreasing. On the other hand, by Lemma 2.6, satisfies (2.14). But this is not possible. Therefore, and there exists such that (2.31) holds. Corollary 2.9 yields satisfying (2.32) with . Therefore, has just one negative local minimum between its first zero and second zero .Step 2. By (2.32) there exists a maximal interval , where . If , then is eventually positive and increasing. On the other hand, by Lemma 2.6, satisfies (2.14). We get a contradiction. Therefore and there exists such that (2.33) holds. Corollary 2.9 yields satisfying (2.34) with . Therefore has just one positive maximum between its second zero and third zero .Step 3. We can continue as in Steps 1 and 2 and get the sequences and of positive local maxima and negative local minima of , respectively. Therefore is oscillatory. Using arguments of the proof of Theorem  3.4 of [18], we get that the sequence is decreasing and the sequence is increasing. In particular, we use (2.5) and define a Lyapunov function by then Consequently, So, sequences and are decreasing and Finally, due to (1.4), the sequence is decreasing and the sequence is increasing. Hence, the sequence of amplitudes is decreasing, as well.

Example 2.11. Consider the problem (1.7), (1.13), where and . In Example 2.4, we have shown that (1.2)–(1.5), (1.8), and (1.9) with , are valid. Since we see that (2.19), (2.20), and (2.29) are satisfied. Therefore, by Theorem 2.10, each damped solution of (2.8), (1.13) with is oscillatory and its amplitudes are decreasing.

Example 2.12. Consider the problem (1.7), (1.13), where then , , We can check that also all remaining assumptions of Theorem 2.10 are satisfied, and this theorem is applicable here.

Assume that does not fulfil (2.20) and (2.29). It occurs, for example, if with for in some neighbourhood of 0, then Theorem 2.10 cannot be applied. Now, we will give another sufficient conditions for the existence of oscillatory solutions. For this purpose, we introduce the following lemmas.

Lemma 2.13. Assume (1.2)–(1.5), (1.8), (1.9), and Let be a solution of the problem (1.7), (1.13) with , then there exists such that

Proof. Assume that such does not exist, then is positive on and, by Lemma 2.6, satisfies (2.14). In view of (1.7) and (1.2), we have on . From (2.45), it follows that there exists such that Motivated by arguments of [27], we divide (1.7) by and integrate it over interval . We get Using the per partes integration, we obtain From (1.8) and (1.9), it follows that there exists such that and therefore From the fact that for (see (2.45)), we have then Multiplying this inequality by , we get and integrating it over , we obtain and therefore, According to (2.53), we have and consequently, Integrating it over , we get From (2.44), it follows that which is a contradiction.