On unbounded solutions of singular IVPs with phi-Laplacian

Electronic Journal of Qualitative Theory of Differential Equations 2017, No. 80, 1–26; https://doi.org/10.14232/ejqtde.2017.1.80 www.math.u-szeged.hu/ejqtde/ On unbounded solutions of singular IVPs with φ-Laplacian Martin RohlederB1,Jana Burkotová1,Lucía López-Somoza2and Jakub Stryja3 1Department of Mathematics, Faculty of Science, Palacký University Olomouc 17. listopadu 12, 771 46 Olomouc, Czech Republic 2Institute of Mathematics, Faculty of Mathematics, University of Santiago de Compostela Lope Gómez de Marzoa, 15782, Santiago de Compostela, Spain 3Department of Mathematics and Descriptive Geometry, VŠB - Technical University Ostrava 17. listopadu 15, 708 33 Ostrava, Czech Republic Received 30 May 2017, appeared 21 November 2017 Communicated by Zuzana Došlá Abstract. The paper deals with a singular nonlinear initial value problem with a φ-Laplacian (p(t)φ(u0(t)))0+p(t)f(φ(u(t))) = 0, t>0, u(0) = u0∈[L0,L],u0(0) = 0. Here, fis a continuous function with three roots φ(L0)<0<φ(L),φ:R→Ris an increasing homeomorphism and function pis positive and increasing on (0, ∞). The problem is singular in the sense that p(0) = 0 and 1/pmay not be integrable in a neighbourhood of the origin. The goal of this paper is to prove the existence of unbounded solutions. The investigation is held in two different ways according to the Lipschitz continuity of functions φ−1and f. The case when those functions are not Lipschitz continuous is more involved that the opposite case and it is managed by means of the lower and upper functions method. In both cases, existence criteria for unbounded solutions are derived. Keywords: second order ODE, time singularity, φ-Laplacian, unbounded solution, escape solution, lower and upper functions method. 2010 Mathematics Subject Classification: 34A12, 34D05, 34C11. 1 Introduction The aim of this paper is to analyse the singular nonlinear equation (p(t)φ(u0(t)))0+p(t)f(φ(u(t))) = 0, t>0, (1.1) BCorresponding author. Email: [email protected] 2M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja with the initial conditions u(0) = u0,u0(0) = 0, u0∈[L0,L]. (1.2) Here we focus our attention on unbounded solutions of problem (1.1), (1.2) and provide sufficient conditions for their existence, while in [6] we discussed the existence and properties of bounded solutions of problem (1.1), (1.2). So, in a way, this paper completes results obtained in [6]. Problem (1.1), (1.2) is investigated under the basic assumptions φ∈C1(R),φ0(x)>0 for x∈(R\{0}), (1.3) φ(R) = R,φ(0) = 0, (1.4) L0<0<L,f(φ(L0)) = f(0) = f(φ(L)) = 0, (1.5) f∈C[φ(L0),∞),x f (x)>0 for x∈((φ(L0),φ(L)) \{0}),f(x)≤0 for x>φ(L), (1.6) p∈C[0, ∞)∩C1(0, ∞),p0(t)>0 for t∈(0, ∞),p(0) = 0. (1.7) As a model example, we can consider problem (1.1), (1.2) with α-Laplacian φ(x) = |x|αsgn x, α≥1, x∈R, and with a three degree polynomial f(x) = x(x−φ(L0))(φ(L)−x),x∈R. For simplicity we can consider function pas a power function p(t) = tβ,β>0, t≥0. Definition 1.1. Let [0, b)⊂[0, ∞)be a maximal interval such that a function u∈C1[0, b) with φ(u0)∈C1(0, b)satisfies equation (1.1) for every t∈(0, b). Then uis called a solution of equation (1.1)on[0, b). If uis a solution of equation (1.1) on [0, ∞), then uis called a solution of equation (1.1). A solution uof equation (1.1) on [0, b)which satisfies the initial conditions (1.2) is called a solution of problem (1.1),(1.2)on [0, b). If uis a solution of problem (1.1), (1.2) on [0, ∞), then uis called a solution of problem (1.1),(1.2). Definition 1.2. Consider a solution of problem (1.1), (1.2) with u0∈(L0,L)and denote usup =sup{u(t):t∈[0, ∞)}. If usup =L, then uis called a homoclinic solution of problem (1.1), (1.2). If usup <L, then uis called a damped solution of problem (1.1), (1.2). Remark 1.3. Assumption (1.5) yields that constant functions u(t)≡L0,u(t)≡0 and u(t)≡L are solutions of problem (1.1), (1.2) on [0, ∞)with u0=L0,u0=0 and u0=L, respectively. If u(0) = 0, then u0cannot be positive on (0, δ)for any δ>0, since then uis positive on (0, δ) and integrating equation (1.1) from 0 to t∈(0, δ), we get, by (1.6), p(t)φ(u0(t)) = −Zt 0p(s)f(φ(u(s)))ds<0, a contradiction. Similarly, u0cannot be negative. Therefore, the solution u(t)≡0 is the unique solution of problem (1.1), (1.2) with u0=0 and clearly, it is a damped solution. Solutions from Definition 1.2 are bounded. Therefore, we are mostly interested in another type of solutions specified in the next definition. Definition 1.4. Let ube a solution of problem (1.1), (1.2) on [0, b), where b∈(0, ∞]. If there exists c∈(0, b)such that u(c) = L,u0(c)>0, (1.8) then uis called an escape solution of problem (1.1), (1.2) on [0, b). On unbounded solutions of singular IVPs with φ-Laplacian 3 A special case of equation (1.1) with φ(u)≡uand p(t) = tn−1,n∈N,n≥2 tn−1u0(t)0+tn−1f(u(t)) = 0, t>0, arises in many areas. For example in the study of phase transition of Van der Waals fluids [11], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [10], in the homogeneous nucleation theory [1], in the relativistic cosmology for description of particles which can be treated as domains in the universe [17], or in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. The above nonlinear equation was replaced with its abstract and more general form p(t)u0(t)0+q(t)f(u(t)) = 0, t>0, which was investigated for p≡qin [20–25] and for p6≡ qin [5,7,26,27]. Other problems without φ-Laplacian close to (1.1), (1.2) can be found in [2–4,13–15] and those with φ-Laplacian in [9,12,16,18,19]. Analytical properties of solutions of problem (1.1), (1.2) with a φ-Laplacian have been already studied in [6] with a focus on existence of bounded solutions on [0, ∞). In more details, the existence of damped solutions was proved for u0∈[¯ B,L]. Some results derived in [6] are also useful here when the existence and properties of unbounded solutions are of interest. Therefore, we recapitulate them in Section 2for the reader’s convenience. The goal of this paper is to find conditions which guarantee the existence of escape solutions of problem (1.1), (1.2), which are unbounded. The analysis of problem (1.1), (1.2) with a general φ-Laplacian includes also φ(x) = |x|αsgn x, for α>1. Let us emphasise that in this case, φ−1(x) = |x|1 αsgn xis not locally Lipschitz continuous. Since φ−1is present in the integral form of (1.1), (1.2) u(t) = u0+Zt 0φ−1−1 p(s)Zs 0p(τ)f(φ(u(τ)))dτds,t≥0, the standard technique based on the Lipschitz property is not applicable here and another approach needs to be developed. Therefore, we distinguish two cases. • In the first case, where functions φ−1and fare Lipschitz continuous, the uniqueness of a solution of problem (1.1), (1.2) is guaranteed. This considerably helps to derive conditions when a sequence of solutions contains an escape solution. • In the second case, functions φ−1and fdo not have to be Lipschitz continuous. The lack of uniqueness causes difficulties and therefore is more challenging. The problems are overcome by means of the lower and upper functions method. Also here sufficient conditions for the existence of escape solutions are derived. Since in general an escape solution needs not be unbounded, criteria for an escape solution to tend to infinity are derived. In this manner, we obtain new existence results for unbounded solutions of problem (1.1), (1.2). The aim of our further research is to analyse the existence of homoclinic solutions. The paper is organised in the following manner: Preliminary results for an auxiliary problem with a bounded nonlinearity are stated in Section 2. Auxiliary lemmas necessary for proofs of the existence of escape solutions of the auxiliary problem are given in Section 3. The 4M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja existence of escape solutions of this problem is further discussed in Section 4. Namely, the first existence result in Section 4is derived by an approach based on the Lipschitz property. The other case without the Lipschitz condition is studied by means of the lower and upper functions method. In Section 5, the criteria for escape solutions of the original problem to be unbounded are proved. The main results about the existence of unbounded solutions with examples are given in Section 6. 2 Preliminary In order to derive the main existence results about unbounded solutions of problem (1.1), (1.2), we first introduce the auxiliary equation with a bounded nonlinearity (p(t)φ(u0(t)))0+p(t)˜ f(φ(u(t))) = 0, t∈(0, ∞), (2.1) where ˜ f(x) = (f(x)for x∈[φ(L0),φ(L)], 0 for x<φ(L0),x>φ(L).(2.2) Since ˜ fis bounded on R, the maximal interval of existence for each solution of problem (2.1), (1.2) is [0, ∞). In this section, we collect preliminary results for solutions of problem (2.1), (1.2) derived in [6]. Properties, asymptotic behaviour and a priori estimates of such solutions are specified in Lemmas 2.1–2.8. The existence and continuous dependence on initial values of solutions is provided in Theorem 2.9 and Theorem 2.10, respectively. Lemma 2.1 (Lemma 2.1 b) in [6]).Let (1.3)–(1.7)hold and let u be a solution of equation (2.1). Assume that there exists a ≥0such that u(a)∈(0, L)and u0(a) = 0. Then u0(t)<0for t ∈(a,θ], where θis the first zero of u on (a,∞).If such θdoes not exist, then u0(t)<0for t ∈(a,∞). Lemma 2.2 (Lemma 2.2 in [6]).Let (1.3)–(1.7)hold and let u be a solution of equation (2.1). Assume that there exists a ≥0such that u(a) = L and u0(a) = 0. a) Let θ>a be the first zero of u on (a,∞). Then there exists a1∈[a,θ)such that u(a1) = L,u0(a1) = 0, 0 ≤u(t)<L,u0(t)<0, t∈(a1,θ]. b) Let u >0on [a,∞)and u 6≡ L on [a,∞). Then there exists a1∈[a,∞)such that u(a1) = L,u0(a1) = 0, 0 <u(t)<L,u0(t)<0, t∈(a1,∞). In both cases, u(t) = L for t ∈[a,a1]. Lemma 2.3 (Lemma 2.6 in [6]).Assume (1.3)–(1.7), lim t→∞ p0(t) p(t)=0, (2.3) and ∃¯ B∈(L0,0):˜ F(¯ B)=˜ F(L),where ˜ F(x) = Zx 0 ˜ f(φ(s)) ds,x∈R. (2.4) Let u be a solution of equation (2.1)and b ≥0and θ>b be such that u(b)∈[¯ B,0),u0(b) = 0, u(θ) = 0, u(t)<0, t∈[b,θ). Then there exists a ∈(θ,∞)such that u0(a) = 0, u0(t)>0, t∈(b,a),u(a)∈(0, L). On unbounded solutions of singular IVPs with φ-Laplacian 5 Lemma 2.4 (Lemma 2.7 in [6]).Assume that (1.3)–(1.7),(2.3)and (2.4)hold. Let u be a solution of equation (2.1)and a ≥0and θ>a be such that u(a)∈(0, L],u0(a) = 0, u(θ) = 0, u(t)>0, t∈[a,θ). Then there exists b ∈(θ,∞)such that u0(b) = 0, u0(t)<0, t∈(a,b),u(b)∈(¯ B,0). Lemma 2.5 (Lemma 2.8 in [6]).Assume that (1.3)–(1.7)and (2.3)hold. Let u be a solution of equation (2.1)and b ≥0be such that u(b)∈(L0,0),u0(b) = 0, u(t)<0, t∈[b,∞). Then lim t→∞u(t) = 0, lim t→∞u0(t) = 0. Lemma 2.6 (Lemma 3.1 in [6]).Assume that (1.3)–(1.7),(2.3)and (2.4)hold. Let u be a solution of problem (2.1),(1.2)with u0∈(L0,¯ B). Let θ>0, a >θbe such that u(θ) = 0, u(t)<0, t∈[0, θ),u0(a) = 0, u0(t)>0, t∈(θ,a). Then u(a)∈(0, L],u0(t)>0, t∈(0, a). Lemma 2.7 (Lemma 3.2 in [6]).Let assumptions (1.3)–(1.7),(2.3)and (2.4)hold. Let u be a solution of problem (2.1),(1.2)with u0∈(L0,0)∪(0, L). Then u0∈[¯ B,0)∪(0, L)⇒¯ B<u(t)<L,t∈(0, ∞), u0∈(L0,¯ B)⇒u0<u(t),t∈(0, ∞). For the following result, we introduce a function ϕ ϕ(t):=1 p(t)Zt 0p(s)ds,t∈(0, T],ϕ(0) = 0. (2.5) This function is continuous on [0, T]and satisfies 0<ϕ(t)≤t,t∈(0, T], lim t→0+ϕ(t) = 0. (2.6) Moreover, we point out that ˜ fis bounded and there exists a constant ˜ M>0 such that |˜ f(x)| ≤ ˜ M,x∈R. (2.7) Lemma 2.8 (Lemma 3.4 in [6]).Assume (1.3)–(1.7). Let u be a solution of problem (2.1),(1.2)with u0∈[L0,L]. The inequality Zβ 0 p0(t) p(t)φ(u0(t))dt≤˜ M(β−ϕ(β)) is valid for every β>0. If moreover (2.3)and (2.4)hold, then there exists ˜ c>0such that |u0(t)| ≤ ˜ c,t∈[0, ∞), for every solution u of (2.1),(1.2)with u0∈(L0,0)∪(0, L). 6M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja The existence of solutions of the auxiliary problem (2.1), (1.2) is proved in [6] by means of the Schauder fixed point theorem. We state this existence result in the next theorem. Theorem 2.9 (Theorem 4.1 in [6]).Assume (1.3)–(1.7). Then, for each u0∈[L0,L], there exists a solution u of problem (2.1),(1.2). The uniqueness of solutions of (2.1), (1.2) follows from the continuous dependence on initial values. This assertion is based on the Lipschitz property, see (2.8) and (2.9). Theorem 2.10 (Theorem 4.3 in [6]).Assume (1.3)–(1.7)and f∈Lip [φ(L0),φ(L)], (2.8) φ−1∈Liploc(R). (2.9) Let uibe a solution of problem (2.1),(1.2)with u0=Bi∈[L0,L], i =1,2. Then, for each β>0, there exists K >0such that ku1−u2kC1[0,β]≤K|B1−B2|. Furthermore, any solution of problem (2.1),(1.2)with u0∈[L0,L]is unique. Remark 2.11. The above lemmas are proved in [6] under the weaker assumption limsup t→∞ p0(t) p(t)<∞ instead of condition (2.3). Similarly, no sign condition of f(x),x/∈[L0,L]is needed in [6] while here we use (1.6). To keep the formulation as simple as possible, we decided to use these additional conditions in formulations of results in this section, whereas the results are proved in [6] without it. 3 Auxiliary results In this section, we provide auxiliary lemmas, which are used in Section 4for proofs of the existence of escape solutions of the auxiliary problem (2.1), (1.2). Note that all solutions of problem (2.1), (1.2) with u0∈[¯ B,L)are damped solutions, see Remark 1.3 and Lemma 2.7. Therefore, we consider only u0∈[L0,¯ B)for investigation of escape solutions of problem (2.1), (1.2). Such solutions can be equivalently characterized as follows. Lemma 3.1. Let (1.3)–(1.7),(2.3)and (2.4)hold and let u be a solution of problem (2.1),(1.2). Then u is an escape solution if and only if sup{u(t):t∈[0, ∞)}>L. (3.1) Proof. Let ufulfils (3.1). According to Definition 1.2,uis not a damped solution and hence, due to Lemma 2.7,u(0)<¯ B<0. Consequently, there exists a maximal c>0 such that u(t)<Lfor t∈[0, c)and u(c) = L,u0(c)≥0. Assume that u0(c) = 0. Using Lemma 2.2 (and in the case of more roots of ualso Lemma 2.3 and Lemma 2.4), we get that sup{u(t):t∈[0, ∞)}=u(c) = L, contrary to (3.1). Therefore, ufulfils (1.8). On the other hand, if uis an escape solution of problem (2.1), (1.2), then (3.1) follows immediately from Definition 1.4. On unbounded solutions of singular IVPs with φ-Laplacian 7 The proofs of the existence of escape solutions are based on Lemma 3.2 and Lemma 3.5. These lemmas are denoted here as Basic lemmas because they are essential for the proof of existence of escape solutions. The Basic lemma I, Lemma 3.2, fully covers the case when the uniqueness of solutions of (2.1), (1.2) is guaranteed. In particular, u≡L0is the unique solution with u0=L0. Therefore, u0=L0is not discussed in the context of escape solutions. The situation is different when (2.8) and (2.9) do not hold, see Basic lemma II, Lemma 3.5. Lemma 3.2 (Basic lemma I).Let (1.3)–(1.7),(2.3)and (2.4)hold. Choose C ∈(L0,¯ B)and a sequence {Bn}∞ n=1⊂(L0,C). Let for each n ∈N, unbe a solution of problem (2.1),(1.2)with u0=Bnand let (0, bn)be the maximal interval such that un(t)<L,u0 n(t)>0, t∈(0, bn). (3.2) Finally, let γn∈(0, bn)be such that un(γn) = C,∀n∈N. (3.3) If the sequence {γn}∞ n=1is unbounded, then the sequence {un}∞ n=1contains an escape solution of problem (2.1),(1.2). Proof. Let the sequence {γn}∞ n=1be unbounded, then there exists a subsequence going to infinity as n→∞. For simplicity, let us denote it by {γn}∞ n=1. Then we have lim n→∞γn=∞,γn<bn,n∈N. Assume on the contrary that for any n∈N,unis not an escape solution of problem (2.1), (1.2). By Lemma 3.1, sup{un(t):t∈[0, ∞)} ≤ L,n∈N. (3.4) STEP 1. Fix n∈Nand consider a solution unof problem (2.1), (1.2) with u0=Bn. First assume that un<0 on [0, ∞). Then, by Lemma 2.1, we get u0 n>0 on (0, ∞), and for bn=∞, we obtain (3.2). In addition, we get by Lemma 2.5 lim t→∞un(t) = 0, lim t→∞u0 n(t) = 0. If we put lim t→∞un(t) =:un(bn), lim t→∞u0 n(t) =:u0 n(bn), we get un(bn) = 0, u0 n(bn) = 0. (3.5) Now we assume that θ>0 is the first zero of un. By Lemma 2.1,u0 n>0 on (0, θ]. (i) Let u0 n>0 on (θ,∞). Then according to (3.4), 0 <un<Lon (θ,∞)and (3.2) is valid for bn=∞. First we prove that lim t→∞un(t) = L, lim t→∞u0 n(t) = 0. Since unis increasing on (0, ∞), then according to (3.4), 0 <un<Lon (0, ∞). We denote lim t→∞un(t) =:`∈(0, L]. 8M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja Since unis a solution of equation (2.1), then φ0(u0 n(t)) u00 n(t) + p0(t) p(t)φ(u0 n(t)) + ˜ f(φ(un(t))) = 0, t∈(0, ∞). (3.6) If we restrict the previous equation to the interval (θ,∞)then, by (1.3)–(1.7), we have that p0(t) p(t)φ(u0 n(t)) >0, ˜ f(φ(un(t))) >0, φ0(u0 n(t)) >0, so we deduce that u00 n(t)<0, t∈(θ,∞). Consequently, u0 nis decreasing on (θ,∞)and so, there must exist limt→∞u0 n(t)≥0. If limt→∞u0 n(t) = a>0, then limt→∞un(t) = ∞, which is a contradiction. Therefore, lim t→∞u0 n(t) = 0. Finally, assume that `∈(0, L). Letting t→∞in (3.6), we get, by (1.4) and (2.3), φ0(0)·lim t→∞u00 n(t) = −˜ f(φ(`)). Since ˜ f(φ(`)) ∈(0, ∞), we get limt→∞u00 n(t)<0, contrary to limt→∞u0 n(t) = 0. Therefore, `=L. Then un(bn) = L,u0 n(bn) = 0. (3.7) (ii) Let a>θbe the first zero of u0 n. By (3.4) we have un(a)≤L. For bn=awe get (3.2) and un(bn)∈(0, L],u0 n(bn) = 0. (3.8) To summarize (3.5), (3.7), (3.8), we see that unfulfils: un(bn)∈[0, L],u0 n(bn) = 0. (3.9) STEP 2. Let nbe fixed. We define En(t):=Zu0 n(t) 0xφ0(x)dx+˜ F(un(t)),t∈(0, bn), and Kn:=sup p0(t) p(t):t∈[γn,bn). Due to (2.3), limn→∞Kn=0. In addition, ∃γn∈[γn,bn):u0 n(γn) = max{u0 n(t):t∈[γn,bn)}. (3.10) Then, by (3.6), the following holds dEn(t) dt=u0 n(t)φ0(u0 n(t)) u00 n(t) + ˜ f(φ(un(t))) u0 n(t) =−p0(t) p(t)φ(u0 n(t)) u0 n(t)<0, t∈(0, bn). On unbounded solutions of singular IVPs with φ-Laplacian 9 Integrating the above equality over (γn,bn)and using (3.2), (3.10), we obtain En(γn)−En(bn) = Zbn γn p0(t) p(t)φ(u0 n(t))u0 n(t)dt≤φ(u0 n(γn)) Zbn γn p0(t) p(t)u0 n(t)dt ≤φ(u0 n(γn))KnZbn γn u0 n(t)dt≤φ(u0 n(γn))Kn(L−C). Hence, we have En(γn)≤En(bn) + φ(u0 n(γn))Kn(L−C). Moreover, from (3.9), we have En(γn)>F(un(γn)) = F(C),En(bn) = F(un(bn)) ≤F(L). This leads to F(C)<En(γn)≤F(L) + φ(u0 n(γn))Kn(L−C). Hence, we derive the estimate F(C)−F(L) L−C 1 Kn <φ(u0 n(γn)). (3.11) STEP 3. We consider a sequence {un}∞ n=1. Since limn→∞Kn=0, we derive from (3.11) that lim n→∞φ(u0 n(γn)) = ∞. (3.12) Using (1.4), we obtain lim n→∞u0 n(γn) = lim n→∞φ−1(φ(u0 n(γn))) = ∞. Since ˜ F≥0 and Enis decreasing on (0, bn), Zu0 n(γn) 0xφ0(x)dx≤En(γn)≤En(γn)≤˜ F(L) + φ(u0 n(γn))Kn(L−C),n∈N therefore, lim n→∞Zu0 n(γn) 0xφ0(x)dx−φ(u0 n(γn))Kn(L−C)≤˜ F(L)<∞. Since lim n→∞u0 n(γn) = ∞, then there exists n0∈Nsuch that u0 n(γn)>1, n≥n0. Therefore, Zun(γn) 0xφ0(x)dx>Zu0 n(γn) 1xφ0(x)dx>Zu0 n(γn) 1φ0(x)dx=φ(u0 n(γn)) −φ(1),n≥n0. By (3.12) and limn→∞Kn=0 we derive lim n→∞Zu0 n(γn) 0xφ0(x)dx−φ(u0 n(γn))Kn(L−C)≥lim n→∞φ(u0 n(γn)) (1−Kn(L−C))−φ(1)=∞. This yields a contradiction. Therefore, the sequence {un}∞ n=1contains an escape solution of problem (2.1), (1.2). 16 M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja Therefore, σ2satisfies conditions (4.5)–(4.7) and so, σ2is an upper function of (2.1), (4.1). STEP 3. Existence of a solution uT: We have found a pair of lower and upper functions which clearly satisfy that σ1(t)≤σ2(t),t∈[0, T]for each T>γ. As a consequence, Theorem 4.5 ensures the existence of a solution uTof problem (2.1), (4.1) such that L0≤uT(t)≤σ2(t),t∈[0, T]. Since σ2(0) = uT(0) = L0,usatisfies (1.2) with u0=L0. Finally, since ˜ f(φ)is bounded on R,uTcan be extended to interval [0, ∞)as a solution of equation (2.1). This classical extension result follows from more general Theorem 11.5 in [14]. The estimate uT>L0on [0, ∞)can be proved in the same way as in the proof of Lemma 3.2 in [6] using Lemma 3.3 instead of Lemmas 2.1 and 2.6. Therefore, uTis a solution of problem (2.1), (1.2) with u0=L0and satisfies (4.11). Theorem 4.7 (Existence of escape solutions of problem (2.1), (1.2) II).Let (1.3)–(1.7),(2.3)and (2.4)hold. Then there exist infinitely many escape solutions of problem (2.1),(1.2)with not necessarily different starting values in [L0,¯ B). Proof. Choose n∈N,C∈(L0,¯ B)and Bn∈(L0,C). By Theorem 2.9, there exists a solution unof problem (2.1), (1.2) with u0=Bn. By Lemma 2.1, there exists a maximal an>0 such that u0 n>0 on (0, an). Since un(0)<0, there exists a maximal ˜ an>0 such that un<Lon [0, ˜ an). If we put bn=min{an,˜ an}, then (3.2) holds. Due to Lemmas 2.1 and 2.5, there exists γn∈(0, bn)such that un(γn) = C. Consider a sequence {Bn}∞ n=1⊂(L0,C). Then we get a sequence {un}∞ n=1of solutions of problem (2.1), (1.2) with u0=Bn, and the corresponding sequence of {γn}∞ n=1. Assume that limn→∞Bn=L0. Now, integrating equation (2.1) we get the equivalent form of problem (2.1), (1.2) for un un(t) = Bn+Zt 0φ−1−1 p(s)Zs 0p(τ)˜ f(φ(un(τ)))dτds,t∈[0, ∞). (4.13) We prove that the sequence {un}∞ n=1is uniformly bounded on [0, β]for all β>0. Indeed, for t∈[0, β], |un(t)|≤|L0|+Zt 0φ−1(˜ Mϕ(s))ds≤ |L0|+Zt 0φ−1(˜ Mβ)ds≤ |L0|+β φ−1(˜ Mβ) =:Kβ, where ϕis defined in (2.5) and ˜ Mis from (2.7). Moreover, as a consequence of Lemma 2.8, we know that the sequence of derivatives {u0 n}∞ n=1is uniformly bounded. Therefore, the sequence {un}∞ n=1is equicontinuous. Therefore, by Arzelà–Ascoli theorem, there exists a subsequence of {un}∞ n=1which converges locally uniformly on [0, ∞)to a continuous function u. To the sake of simplicity we denote this subsequence also as {un}∞ n=1. In particular, if we take the limit when tgoes to infinity on equation (4.13), since the convergence is locally uniform, we obtain that usatisfies the following u(t) = L0+Zt 0φ−1−1 p(s)Zs 0p(τ)˜ f(φ(u(τ)))dτds,t∈[0, ∞), and therefore, uis a solution of problem (2.1), (1.2) for u0=L0. Now, we distinguish three different cases. On unbounded solutions of singular IVPs with φ-Laplacian 17 (i) u≡L0: In this case, limn→∞γn=∞and the sequence {γn}∞ n=1is unbounded. By Lemma 3.2 there exists n0∈Nsuch that un0is an escape solution of problem (2.1), (1.2). We have un0(0) = Bn0>L0. Now consider the unbounded sequence {γn}∞ n=n0+1. By Lemma 3.2 there exists n1∈Nsuch that un1is an escape solution of problem (2.1), (1.2) with un1(0) = Bn1>L0. We repeat this procedure and we obtain the sequence {unk}∞ k=0of escape solutions of problem (2.1), (1.2) with starting values in (L0,¯ B). (ii) u6≡ L0is not an escape solution: In this case, we define ˜ Bn=L0for all n∈Nand consider γdefined in (4.12). Now, we can take an unbounded sequence {˜ γn}∞ n=1such that ˜ γn>γfor all n∈N. By Lemma 4.6, for all n∈Nthere exists a solution ˜ unof problem (2.1), (1.2) with u0=˜ Bnsuch that ˜ un(˜ γn) = C,˜ un(t)≥L0,t∈[0, ∞). Therefore, we have a sequence of solutions {˜ un}∞ n=1in the conditions of Lemma 3.5 and so, this sequence contains an escape solution ˜ un0of (2.1), (1.2) with u0=L0. As in the previous case, we could consider now the unbounded sequence {˜ γn}∞ n=n0+1and repeat the procedure from (i). This way we obtain a sequence {˜ unk}∞ k=0of escape solutions of problem (2.1), (1.2) with u0=L0. (iii) u6≡ L0is an escape solution: In this case, we can argue as in (ii)and we also obtain a sequence {˜ unk}∞ k=0of escape solutions of problem (2.1), (1.2) with u0=L0. Moreover, in this case, since the sequence {un}∞ n=0converges locally uniformly to an escape solution of (2.1), (1.2), there must exist some n0such that unis also an escape solution for all n≥n0. As a consequence we also obtain a sequence {un}∞ n=n0of escape solutions of problem (2.1), (1.2) with starting values in (L0,¯ B). 5 Unbounded solutions In this section, we discuss the original problem (1.1), (1.2) and provide conditions which guarantee that an escape solution of (1.1), (1.2) is unbounded. Note that solutions of the original problem (1.1), (1.2) and solutions of the auxiliary problem (2.1), (1.2) are related in the following way (when (1.3)–(1.7), (2.3) and (2.4) are assumed): Each solution of (2.1), (1.2) which is not an escape solution, is a bounded solution of the original problem (1.1), (1.2) in [0, ∞). This results from Lemma 2.7 and Lemma 3.1, where such solutions of (2.1), (1.2) satisfy L0≤u(t)≤L,t∈[0, ∞) and, due to (2.2), ˜ f(φ(u(t))) = f(φ(u(t))),t∈[0, ∞). If uis an escape solution of the auxiliary problem (2.1), (1.2), i.e. ∃c∈(0, ∞):u(t)∈[L0,L),t∈[0, c),u(c) = L,u0(c)>0, (5.1) 18 M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja then ufulfils at once the auxiliary equation (2.1) and the original equation (1.1) on [0, c]. The restriction of uon [0, c]can be extended as an escape solution of problem (1.1), (1.2) on some maximal interval [0, b). Therefore, we search for unbounded solutions of (1.1), (1.2) in the set of escape solutions of (1.1), (1.2) on [0, b). Since in general, an escape solution uof (1.1), (1.2) on [0, b)need not to be unbounded, we derive criteria for uto tend to infinity. Lemma 5.1. Assume that (1.3)–(1.7)hold. Let u be an escape solution of problem (1.1),(1.2)on [0, b). Then u(t)>L,u0(t)>0, t∈(c,b), (5.2) where c is from (5.1). If b <∞, then lim t→b−u(t) = ∞. Proof. Let ube an escape solution of problem (1.1), (1.2) on [0, b). Then (5.1) holds. Assume that there exists c1>csuch that u0(c1) = 0, u(t)>L,u0(t)>0 for t∈(c,c1). Integrating equation (1.1) over [c,c1], dividing by p(t)and using (1.3), (1.4), (1.6), (1.7), we get φ(u0(t)) = p(c)φ(u0(c)) p(t)−1 p(t)Zt cp(s)f(φ(u(s)))ds>0, t∈[c,c1], contrary to u0(c1) = 0. Hence, u(t)>Land u0(t)>0 for t∈(c,b)which yields (5.2). Let b<∞. Since [0, b)is the maximal interval, where the solution uis defined, ucannot be extended behind b. Therefore, (5.2) gives limt→b−u(t) = ∞and thus, the solution uis unbounded. Since all escape solutions of (2.1), (1.2) on [0, b)which cannot be extended on the halfline [0, ∞)are naturally unbounded, we continue our investigation about unboundedness of escape solutions defined on [0, ∞). Theorem 5.2. Assume (1.3)–(1.7)hold and let lim t→∞p(t)<∞. (5.3) Let u be an escape solution of problem (1.1),(1.2). Then lim t→∞u(t) = ∞. (5.4) Proof. Let ube an escape solution of problem (1.1), (1.2). Lemma 5.1 gives (5.2) with b=∞ and so, there exists limt→∞u(t)∈(L,∞]. Due to (1.3), (1.4), (1.7) and (5.1), p(c)φ(u0(c)) =: c0∈(0, ∞). Integrate equation (1.1) from cto t>cand get, by (1.6) and (1.7), u(t) = L+Zt cφ−1c0 p(s)−1 p(s)Zs cp(τ)f(φ(u(τ)))dτds>Zt cφ−1c0 p(s)ds,t∈(c,∞). Conditions (1.7) and (5.3) give lims→∞c0 p(s)∈(0, ∞)and, by (1.3) and (1.4), Z∞ 1φ−1c0 p(s)ds=∞. Therefore, lim t→∞u(t)≥Z∞ cφ−1c0 p(s)ds=∞, which gives (5.4). On unbounded solutions of singular IVPs with φ-Laplacian 19 Theorem 5.3. Assume (1.3)–(1.7),(2.3)and f(x)<0for x >φ(L). (5.5) Let u be an escape solution of problem (1.1),(1.2). Then (5.4)holds. Proof. Let ube an escape solution of problem (1.1), (1.2). According to Lemma 5.1,u0>0 on (c,∞)and hence, there exists limt→∞u(t)∈(L,∞]. Assume on the contrary that lim t→∞u(t) =:A∈(L,∞). (5.6) STEP 1. We prove that u0is bounded. Assume that u0is unbounded. Then there exists a sequence {tn}∞ n=1such that limn→∞tn=∞, limn→∞u0(tn) = ∞. The next approach is similar to the proof of Lemma 2.8 in [6]. Equation (1.1) has an equivalent form φ0(u0(t))u00(t) + p0(t) p(t)φ(u0(t)) + f(φ(u(t))) = 0, t∈(0, ∞). (5.7) Choose n∈N. Multiplying this equation by u0and integrating it from cto t>c, we obtain for t=tnthat ψ1(tn) + ψ2(tn) + ψ3(tn) = 0, tn∈[c,∞), (5.8) where ψ1(tn) = Zu0(tn) u0(c)xφ0(x)dx,ψ2(tn) = Ztn c p0(s) p(s)φ(u0(s))u0(s)ds,ψ3(tn) = Zu(tn) Lf(φ(x)) dx. Then ψ3(tn) = F(u(tn)) −F(L), where F(x):=Rx 0f(φ(s)) ds,x∈R. Due to (1.3), (1.4) and (5.5), F(x)is decreasing for x>φ(L). Since uis increasing on (c,∞),F(u(tn)) is decreasing for tn∈(c,∞)and limn→∞F(u(tn)) = F(A). According to (5.6), lim n→∞ψ3(tn)∈(−∞,0). By (1.3), (1.4) and (1.7), lim n→∞ψ1(tn) = ∞, lim n→∞ψ2(tn)>0. Hence, letting n→∞in (5.8), we obtain 0=lim n→∞(ψ1(tn) + ψ2(tn) + ψ3(tn)) = ∞, a contradiction. So, u0is bounded. STEP 2. We prove (5.4). Since u0is bounded, letting t→∞in (5.7) and using (2.3), (5.5) and (5.6), we get lim t→∞φ0(u0(t))u00(t) = −f(φ(A)) >0. Since φ0(u0(t)) >0 for t>c, there exists τ>csuch that u00(t)>0 for t≥τ. Therefore, u0is increasing on [τ,∞)and there exists limt→∞u0(t)>0, which contradicts limt→∞u(t) = A< ∞. Thus, (5.4) is valid. Remark 5.4. The proof of Theorem 5.3 yields that if a solution uof problem (1.1), (1.2) satisfies limt→∞u(t) =:A∈(L,∞), then f(φ(A)) = 0, which is equivalent with the fact that u(t)≡A is a solution of equation (1.1). 20 M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja For f≡0 on (φ(L),∞), we are able to find necessary and sufficient condition for the unboundedness of escape solutions of problem (1.1), (1.2). Theorem 5.5. Assume (1.3)–(1.7), f(x)≡0for x >φ(L), (5.9) and φ(x) = xα,x∈(0, ∞),α≥1. (5.10) Let u be an escape solution of problem (1.1),(1.2). Then lim t→∞u(t) = ∞⇐⇒ Z∞ 1φ−11 p(s)ds=∞. (5.11) If we replace condition (5.10)by φ(ab)≤φ(a)φ(b),a,b∈(0, ∞), (5.12) then (5.4)holds if Z∞ 1φ−11 p(s)ds=∞. (5.13) Proof. Let ube an escape solution of problem (1.1), (1.2). According to Lemma 5.1,u0>0 on (c,∞). Then there exists t0>csuch that u(t0)>L,u0(t)>0 for t∈[t0,∞). Therefore, there exists limt→∞u(t)∈(L,∞]. By (5.10), φ−1(ab) = φ−1(a)φ−1(b)for a,b∈(0, ∞). Due to (1.3), (1.4), (1.7) and (5.9), p(t0)φ(u0(t0)) =:c0∈(0, ∞),f(φ(u(t))) = 0 for t∈[t0,∞). Thus, integrating equation (1.1) from t0to t>t0, we get u(t) =u(t0) + Zt t0 φ−1c0 p(s)ds=u(t0) +φ−1(c0)Zt 1φ−11 p(s)ds−Zt0 1φ−11 p(s)ds,t∈(t0,∞). Letting t→∞here, we get (5.11). Let us consider (5.12) instead of (5.10) and assume (5.13). Then we continue analogously and obtain φ−1(a)φ−1(b) = φ−1(φ(φ−1(a)φ−1(b))) ≤φ−1(φ(φ−1(a))φ(φ−1(b))) = φ−1(ab),a,b∈(0, ∞), u(t) = u(t0) + Zt t0 φ−1c0 p(s)ds≥u(t0) +φ−1(c0)Zt 1φ−11 p(s)ds−Zt0 1φ−11 p(s)ds,t∈(t0,∞). We let t→∞here and obtain, by (5.13), that (5.4) holds. On unbounded solutions of singular IVPs with φ-Laplacian 21 6 Main results and examples In this section, we first present the existence results about unbounded solutions of the original problem (1.1), (1.2) in the case when φ−1and fare Lipschitz continuous, see Theorems 6.1, 6.3 and 6.5. Each of these theorems is afterwards illustrated by an example which is chosen in such a way that only this theorem is applicable, while none of the remaining two theorems can be used for this example. Then, in Theorems 6.7,6.9 and 6.11, we present the main existence results about unbounded solutions of the original problem (1.1), (1.2) provided φ−1and fdo not need to be Lipschitz continuous. The illustration by examples is done as in the previous case and shows that none of these theorems is included in any of two remaining ones. In the whole section, we assume that (due to Definition 1.1) for each n∈N,[0, bn)⊂ [0, ∞)is a maximal interval such that a function unsatisfies equation (1.1) for every t∈(0, bn). Theorem 6.1. Assume that (1.3)–(1.7),(2.3),(2.4),(2.8),(2.9)and (5.3)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with different starting values in (L0,¯ B), n ∈N. Proof. By Theorem 4.1, there exist infinitely many escape solutions unof problem (2.1), (1.2) with starting values in (L0,¯ B). Let us choose n∈N. Then ∃cn∈(0, ∞):un(t)∈(L0,L),t∈[0, cn),un(cn) = L,u0 n(cn)>0. Consider restriction of unto [0, cn]. Then there exists bn>cnsuch that uncan be extended as a solution of problem (1.1), (1.2) on [0, bn). If bn<∞, then, due to Lemma 5.1, lim t→b− n un(t) = ∞, so unis unbounded. If bn=∞, then Theorem 5.2 yields lim t→∞un(t) = ∞, that is unis unbounded, as well. Example 6.2. Consider problem (1.1), (1.2) with φ(x) = sinh x=ex−e−x 2,x∈R, f(x) = (x(x+sinh4)(sinh1 −x)for x∈[−sinh 4,sinh1], cos(x−sinh 1)−1 for x>sinh1, p(t) = arctan tor p(t) = tanh t=et−e−t et+e−t,t∈[0, ∞). Here L0=−4, L=1, φ−1(x) = argsinh x=ln x+√x2+1. These functions psatisfy (1.7), (5.3) and lim t→∞ (arctan t)0 arctan t=lim t→∞ 1 t2+1 arctan t=0, lim t→∞ (tanh t)0 tanh t=lim t→∞ 1 cosh2t tanh t=0, 22 M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja that is (2.3) holds, as well. Functions φand ffulfil (1.3)–(1.6). Moreover, 0 <L<−L0,φis odd and ˜ F(L0) = Z−4 0φ(s)(φ(s) + sinh4) (sinh1 −φ(s))ds =Z4 0φ(s)(sinh4 −φ(s)) (sinh1 +φ(s))ds>Z1 0φ(s)(sinh4 −φ(s)) (sinh1 +φ(s))ds >Z1 0φ(s)(φ(s) + sinh4) (sinh1 −φ(s))ds=˜ F(L), thus, (2.4) holds. Since fand φ−1are Lipschitz continuous, conditions (2.8) and (2.9) are valid, too. We have fulfilled all assumptions of Theorem 6.1. Since fhas isolated zeros on (sinh 1, ∞), we cannot use Theorem 6.3 and 6.5 here. In the same way as in the proof of Theorem 6.1, we can prove the following Theorems 6.3 or 6.5, if we use in the proof Theorems 5.3 or 5.5, respectively, instead of Theorem 5.2. Theorem 6.3. Let (1.3)–(1.7),(2.3),(2.4),(2.8),(2.9)and (5.5)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with different starting values in (L0,¯ B), n∈N. Example 6.4. Let us consider problem (1.1), (1.2) with φ(x) = ln(|x|+1)sgn x,x∈R, f(x) = x(x+ln 4)(ln 2 −x),x∈[−ln 4, ∞), p(t) = tβ,β>0, t∈[0, ∞). Here L0=−3, L=1, φ−1(x) = e|x|−1sgn x. We can easily check that φ,fand psatisfy (1.3)–(1.7), (2.3) and (5.5). In addition, 0 <L<−L0,φis odd and we can show similarly as in Example 6.2 that (2.4) holds. The Lipschitz continuity of fand φ−1yields (2.8) and (2.9). Thus, we can apply Theorem 6.3 here. Since limt→∞tβ=∞and f(x)<0 for x>ln2, we cannot use either Theorem 6.1 or Theorem 6.5. Theorem 6.5. Assume that (1.3)–(1.7),(2.3),(2.4),(2.8),(2.9),(5.9),(5.12)and (5.13)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with different starting values in (L0,¯ B), n ∈N. Example 6.6. Consider problem (1.1), (1.2) with φ(x) = x,x∈R, p(t) = √t,t∈[0, ∞), f(x) = (x3(x−φ(L0))(φ(L)−x)for x∈[φ(L0),φ(L)], 0 for x>φ(L),0<L<−L0. Functions φ,f,pand φ−1(x) = xsatisfy (1.3)–(1.7), (2.3), (2.8), (2.9), (5.9), (5.10) and consequently, (5.12). Since f(φ(x)) = f(x)and L<−L0, we have ˜ F(L)<˜ F(L0)and (2.4) holds. In addition, Z∞ 1φ−11 p(s)ds=Z∞ 1 1 √sds=∞, which yields (5.13). We have verified all assumptions of Theorem 6.5. Since limt→∞√t=∞ and f(x) = 0 for x>φ(L), we cannot use either Theorem 6.1 or Theorem 6.3. On unbounded solutions of singular IVPs with φ-Laplacian 23 Now, applying Theorem 4.7 instead of Theorem 4.1, we get as before the existence results about unbounded solutions in the case when φ−1and fdo not have to be Lipschitz continuous. Theorem 6.7. Let (1.3)–(1.7),(2.3),(2.4)and (5.3)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with not necessarily different starting values in [L0,¯ B), n∈N. Example 6.8. Let us consider problem (1.1), (1.2) with φ(x) = |x|αsgn x,α>1, x∈R, f(x) =        p|x|sgn x(x−φ(L0))(φ(L)−x)for x∈[φ(L0),φ(L)], (φ(L)−x)(φ(2L)−x)for x∈(φ(L),φ(2L)), 0 for x≥φ(2L), 0<L<−L0, p(t) = arctan tor p(t) = tanh t=et−e−t et+e−t,t∈[0, ∞). According to Example 6.2, functions psatisfy (1.7), (2.3) and (5.3). Functions φand ffulfil (1.3)–(1.6). Since fis continuous, 0 <L<−L0and φis a continuous and odd function, (2.4) holds, too. We have verified all assumptions of Theorem 6.7. The form of fimplies that neither Theorem 6.9 nor Theorem 6.11 can be applied. Theorem 6.9. Assume that (1.3)–(1.7),(2.3),(2.4)and (5.5)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with not necessarily different starting values in [L0,¯ B), n ∈N. Example 6.10. Consider problem (1.1), (1.2) with φ(x) = x3,x∈R, f(x) = 3 √x(x+8)(1−x),x∈[−8, ∞), p(t) = tβ,β>0, t∈[0, ∞). Here L0=−2, L=1, φ−1(x) = 3 √x. It is easy to see that φ,fand pfulfil (1.3)–(1.7), (2.3) and (5.5). Further, ˜ F(L0) = Z−2 0ss3+81−s3ds=144 5,˜ F(L) = Z1 0ss3+81−s3ds=99 40 . So, ˜ F(L0)>˜ F(L)which yields (2.4). Therefore, we can apply Theorem 6.9 here. Since limt→∞tβ=∞and f(x)<0 for x>1, we cannot use either Theorem 6.7 or Theorem 6.11. Theorem 6.11. Let (1.3)–(1.7),(2.3),(2.4),(5.9),(5.12)and (5.13)hold. Then there exist infinitely many unbounded solutions unof problem (1.1),(1.2)on [0, bn)with not necessarily different starting values in [L0,¯ B), n ∈N. Example 6.12. Let us consider problem (1.1), (1.2) with φ(x) = |x|αsgn x,α>1, x∈R, p(t) = tβ,β∈(0, α],t∈[0, ∞), f(x) = (3 √x(x−φ(L0))(φ(L)−x)for x∈[φ(L0),φ(L)], 0 for x>φ(L),0<L<−L0. 24 M. Rohleder, J. Burkotová, L. López-Somoza and J. Stryja Functions φ,fand psatisfy (1.3)–(1.7), (2.3), (5.9), (5.10) and consequently, (5.12). Moreover, 0<L<−L0and φis odd function which yields (2.4). Further, φ−1(x) = x1 αfor x>0, Z∞ 1φ−11 p(s)ds=Z∞ 1s−β αds=∞, that is (5.13) holds and we have verified all assumptions of Theorem 6.11. Since limt→∞tβ=∞ and f(x) = 0 for x>φ(L), neither Theorem 6.7 nor Theorem 6.9 is applicable. It si clear that every unbounded solution of problem (1.1), (1.2) is an escape solution. According to the proofs of above theorems, we can formulate also the reverse assertion. Corollary 6.13. Assume all assumptions of Theorem 6.1 or 6.3 or 6.5 or 6.7 or 6.9 or 6.11. Then each escape solution of problem (1.1),(1.2)is unbounded. In this paper we discuss the existence of unbounded solutions of the singular nonlinear initial value problem (1.1), (1.2) with a φ-Laplacian. In the case when functions fand φ−1are Lipschitz continuous, a sequence of escape solutions with different initial values in (L0,¯ B)is obtained. The basis of the proof is a sequence of solutions which converge locally uniformly to a solution uwith u0=L0. By virtue of uniqueness u≡L0. This is not guaranteed in the other case when such sequence converging to the constant solution u≡L0might not exist. Therefore we would like to point out the approach without assuming the Lipschitz property of data functions. In this situation, the investigation is not straightforward and requires some efficient idea about how to deal with difficulties caused by the lack of uniqueness. In contrast to the case with Lipschitz data functions, the set of escape solutions with u0∈(L0,¯ B)might be empty and all escape solutions could start at u0=L0. Therefore we cannot just follow the method used in the first case. The technique used here is the method of lower and upper functions which is applied to a sequence of related boundary value problems. The sequence of obtained solutions contains an escape solution, in particular infinitely many escape solutions. These solutions are under suitable conditions unbounded. In this manner, we prove the existence of unbounded solutions of the investigated problem. Acknowledgement The first two authors gratefully acknowledge support received from the grant No. 14-06958S of the Grant Agency of the Czech Republic. The third author was partially supported by Xunta de Galicia (Spain), project EM2014/032, AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P., FPU scholarship (Ministerio de Educación, Cultura y Deporte, Spain) and a Fundación Barrié research stay scholarship. This paper was mostly written during a stay of Lucía López-Somoza in Olomouc. Lucía López-Somoza is very grateful to the members of the Department of Mathematics, Faculty of Science, Palacký University for their kind and hospitality. References [1] F. F. Abraham,Homogeneous nucleation theory, Academic Press, New York, 1974. On unbounded solutions of singular IVPs with φ-Laplacian 25 [2] R. P. Agarwal, D. O’Regan,Infinite interval problems for differential, difference and integral equations, Kluwer, Dordrecht 2001. MR1845855;https://doi.org/10.1007/ 978-94-010-0718-4 [3] H. Berestycki, P. L. Lions, L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in Rn,Indiana Math. Univ. J. 30(1981), 141–157. MR0600039;https://doi.org/10.1512/iumj.1981.30.30012 [4] D. Bonheure, J. M. Gomes, L. Sanchez, Positive solutions of a second-order singular ordinary differential equation, Nonlinear Anal. 61(2005), 1383–1399. MR2135816;https: //doi.org/10.1016/j.na.2005.02.029 [5] J. Burkotová, M. Hubner, I. Rach ˚ unková, E. B. Weinmüller, Asymptotic properties of Kneser solutions to nonlinear second order ODEs with regularly varying coefficients, J. Appl. Math. Comp. 274(2015), 65–82. MR3433115;https://doi.org/10.1016/j.amc. 2015.10.074 [6] J. Burkotová, I. Rach ˚ unková M. Rohleder, J. Stryja, Existence and uniqueness of damped solutions of singular IVPs with φ-Laplacian, Electron. J. Qual. Theory Differ. Equ. 2016, No. 121, 1–28 MR3592201;https://doi.org/10.14232/ejqtde.2016.1.121 [7] J. Burkotová, M. Rohleder, J. Stryja, On the existence and properties of three types of solutions of singular IVPs, Electron. J. Qual. Theory Differ. Equ. 2015, No. 29, 1–25. MR3353200;https://doi.org/10.14232/ejqtde.2015.1.29 [8] G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Physics 5(1965), 1252–1254. MR0174304;https://doi.org/10.1063/1. 1704233 [9] Z. Došlá, M. Marini, S. Matucci, A boundary value problem on a half-line for differential equations with indefinite weight, Commun. Appl. Anal. 15(2011), 341–352. MR2867356 [10] P. C. Fife,Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, Vol. 28, Springer-Verlag, Berlin–New York, 1979. MR0527914;https://doi.org/ 10.1007/978-3-642-93111-6 [11] H. Gouin, G. Rotoli, An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids, Mech. Research Communic. 24(1997), 255–260. https://doi.org/10.1016/S0093-6413(97)00022-0 [12] J. Jaroš, T. Kusano, J. V. Manojlovi´ c, Asymptotic analysis of positive solutions of generalized Emden–Fowler differential equations in the framework of regular variation, Cent. Eur. J. Mathc. 11(2013), No. 12, 2215–2233. MR3111718;https://doi.org/10.2478/ s11533-013-0306-9 [13] I. T. Kiguradze,Some singular boundary value problems for ordinary differential equations, ITU, Tbilisi, 1975. MR0499402 [14] I. T. Kiguradze, T. A. Chanturia,Asymptotic properties of solutions of nonautonomous ordinary differential equations, Kluver Academic, Dordrecht, 1993. MR1220223;https: //doi.org/10.1007/978-94-011-1808-8