Analysis and numerical approximation of singular boundary value problems

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Maria Lu´

ısa Ribeiro dos Santos Morgado

Analysis and Numerical

Approximation of Singular Boundary

Value Problems

Departamento de Matem´

atica

Universidade de Tr´

as-os-Montes e Alto Douro

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Maria Lu´

ısa Ribeiro dos Santos Morgado

Analysis and Numerical

Approximation of Singular Boundary

Value Problems

Disserta¸cão de Doutoramento submetida à Universidade de Trás-os-Montes e Alto Douro para a obten¸cão do Grau de Doutor em Matemática Aplicada.

Departamento de Matem´

atica

Universidade de Tr´

as-os-Montes e Alto Douro

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Disserta¸c˜ao realizada sob a supervis˜ao do Prof. Dr. Pedro Miguel Rita da Trindade e Lima,

Professor Auxiliar com agrega¸c˜ao do Departamento de Matem´atica do

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Agradecimentos

Todo o trabalho por mim desenvolvido nestes últimos anos não teria sido poss´ıvel sem o apoio financeiro e burocrático de algumas institui¸cões, e sem o incentivo de familiares e amigos. A todos, gostaria de deixar aqui os meus sinceros agradecimentos:

Ao Prof. Dr. Pedro Lima, meu orientador cient´ıfico, pelo acompanhamento prestado e pela disponibilidade sempre mantida ao longo da elabora¸c˜ao desta disserta¸c˜ao;

`

A Doutora Nadezhda Konyukhova, pelos esclarecimentos prestados e por muito gen-tilmente ter cedido os seus trabalhos;

Ao Magn´ıfico Reitor da Universidade de Trás-os-Montes e Alto Douro e à própria institui¸cão em geral; ao departamento de Matemática, em particular à coordena¸cão;

Ao Cemat, minha institui¸c˜ao de acolhimento;

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A Funda¸c˜ao para a Ciˆencia e Tecnologia pelo projecto POCTI/MAT/45700/2002 e pela bolsa de Doutoramento SFRH/BD/31513/2006;

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A minha fam´ılia, em especial ao Lu´ıs, `a Filipa e `a Cristiana a quem dedico este trabalho.

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Resumo Alargado

Nos modelos matemáticos de muitos problemas da engenharia, f´ısica, qu´ımica, biolo-gia e outras ciências surgem frequentemente problemas de valores de fronteira singu-lares (abreviadamente PVFs) para equa¸cões diferenciais ordinárias de segunda ordem não lineares. O estudo destes problemas levou ao desenvolvimento de diferentes métodos anal´ıticos e numéricos que permitiram obter resultados sobre: 1) existência e unicidade de solu¸cão, 2) comportamento da solu¸cão numa vizinhan¸ca dos pontos singulares, 3) aprox-ima¸cão numérica da solu¸cão. Neste trabalho estamos particularmente interessados nos dois últimos pontos. A abordagem aqui seguida consiste no estudo de problemas sin-gulares de Cauchy, permitindo assim obter fam´ılias uniparamétricas de solu¸cões (sob a forma de séries assimptóticas ou convergentes) numa vizinhan¸ca das singularidades. Estas fam´ılias são depois usadas na aproxima¸cão numérica da solu¸cão dos problemas de valores de fronteira, ajustando o parâmetro dessas fam´ılias através de algoritmos de shooting.

A motiva¸cão para o trabalho apresentado no cap´ıtulo 2 vem, parcialmente, do seguinte modelo com aplica¸cões a vários problemas de engenharia, meteorologia e oceanografia:

g′′_{(t) = −tg}−1λ(t), 0 < t < 1, λ > 0 (1)

g′_{(0) = 0,} ₍₂₎

g(1) = 0. (3)

No trabalho [29], os autores determinaram fam´ılias uniparamétricas de solu¸cões descrevendo o comportamento da solu¸cão do problema de Cauchy (1), (3), na vizinhan¸ca do ponto sin-gular t = 1, e com base nestas fam´ılias construiram um algoritmo de shooting para a aproxima¸cão numérica da solu¸cão do PVF.

Na primeira seçcão do cap´ıtulo 2 continua-se a investiga¸cão iniciada em [29]. Em particular, generalizam-se os resultados a´ı obtidos a duas classes de PVFs associados à equa¸cão de Emden-Fowler:

g′′_{(u) = au}σ_gn_{(u) ,} _{0 < u < u}

0, (4)

onde σ, n < 0, a < 0 e u0 > 0 s˜ao n´umeros reais. A primeira classe de problemas tem

como condi¸c˜oes de fronteira:

g′(0) = g (u0) = lim u→u−

0

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e a segunda g (0) = lim u→0+ug ′_{(u) = g (u} 0) = lim u→u− 0 [(u0− u) g′(u)] = 0.

Na segunda seçcão do cap´ıtulo 2 consideramos uma classe mais vasta de problemas, considerando em vez da equa¸cão (4), a equa¸cão generalizada de Emden-Fowler:

(|g′(u)|m−2g′(u))′ = auσgn(u) , 0 < u < u0, (5)

na qual n < 0, a < 0, m > 1, u0 > 0 e σ s˜ao reais. Note-se que quando m = 2, a

equa¸cão (5) reduz-se à equa¸cão (4). O operador diferencial no primeiro membro de (5) é o chamado operador m-laplaciano unidimensional (que coincide com o operador laplaciano clássico quando m = 2), e tem várias aplica¸cões, por exemplo, em mecânica de fluidos e termodinâmica. A principal diferen¸ca entre os casos m = 2 e m 6= 2, é que neste último uma outra singularidade pode surgir, num ponto que a priori não é conhecido, no qual a primeira derivada da solu¸cão se anula.

Para cada classe de problemas estudada no cap´ıtulo 2, constru´ımos sub e super solu¸cões, e em alguns casos especiais, obtemos uma fórmula para a solu¸cão exacta, o que nos permite testar a eficiência dos algoritmos numéricos.

Enquanto que o cap´ıtulo 2 é dedicado à análise de PVFs em intervalos limitados e definidos à partida, no cap´ıtulo 3 estudamos algumas classes de problemas de fronteira livre. Neste tipo de problemas que surge em variadas áreas, nomeadamente, em f´ısica do plasma [43] uma das condi¸cões de fronteira é imposta num ponto desconhecido à partida, no qual a solu¸cão e a sua primeira derivada se anulam.

A classe de problemas mais geral estudada neste cap´ıtulo é a seguinte: pretende-se determinar um real M > 0 e uma solu¸cão positiva da equa¸cão

(|y′|m−2y′)′+N − 1 x |y

′_|m−2_y′_{+ f (y) = 0,} _{0 < x < +∞,}

onde N ≥ 2, m > 1 e f(y) = ayq_−byp_{, p < q, a, b > 0, que satisfa¸ca as seguintes condi¸c˜oes}

de fronteira

y′_{(0) = 0,} _{y(M ) = y}′_{(M ) = 0.}

Adaptamos a este tipo de problemas os algoritmos de shooting apresentados no cap´ıtulo anterior e apresentamos ainda um esquema de diferen¸cas finitas.

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ix No cap´ıtulo 4 analisamos PVFs para equa¸cões diferenciais ordinárias em dom´ınios não limitados. Este tipo de problemas surge, por exemplo, em hidrodinâmica [36] e em cosmologia [30]. A análise de modelos semelhantes segundo a abordagem aqui seguida foi iniciada em [36] e é aqui continuada, estudando a seguinte classe de problemas:

y′′_{(r) +}N −1

r y′(r) = c(r)f (y), 0 < r < +∞ (6)

y′_{(0) = 0,} ₍₇₎

lim

r→+∞y(r) = 0, (8)

onde c(r) é uma fun¸cão cont´ınua e limitada em [0, +∞[ e f(y) = y(ξ − y)(y − ξ − 1), 0 < ξ < 1. Os resultados de [36] foram generalizados para o caso em que a fun¸cão c(r) é não constante; por outro lado, considerou-se também o caso f (y) = yq_{− y, onde q > 1 é}

um inteiro, modelo que descreve a distribui¸cão de pressão num fluido de Van der Walls. Finalmente, no último cap´ıtulo, apresentamos as conclusões e alguns problemas em aberto, que serão tema de trabalho futuro.

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Abstract

In recent decades, many works have been devoted to the analysis of singular boundary value problems and many different techniques have been used or developed in order to deal with three main questions: existence and uniqueness of solutions, behavior of the solution in the neighborhood of the singular points and its numerical approximation. In this dissertation we focus our attention on the two last questions, for some classes of singular boundary value problems for nonlinear second order ODEs. Our approach is based on the analysis of the asymptotic behavior of solutions in the neighborhood of singular points. Based on this analysis we obtain expansions of the one-parameter family of solutions (in the form of convergent or asymptotic series), satisfying certain boundary conditions at the singularity, which are used to compute approximate solutions near this singular point. In this way we obtain regular Cauchy problems which are solved by standard methods. Finally, the needed parameter is adjusted using a shooting algorithm.

In chapter 2 this approach is applied to boundary value problems over bounded do-mains, where we investigate two classes of problems: a boundary value problem with two singular endpoints, which was first studied by Taliaferro, [51], and a boundary value problem involving the one-dimensional m-laplacian which, besides the singularities at the endpoints of the interval, has also a singularity at an interior point (whose localization is not known) where the first derivative of the solution vanishes.

In chapter 3 we deal with singular free boundary problems. Such problems arise for example in plasma physics, [43]. In this case, one of the boundary conditions is imposed at a point which is not known, where the solution and its first derivative vanish. We introduce a shooting algorithm and also a finite difference scheme and the numerical results obtained by the two methods are compared. Besides the problems mentioned previously, we also investigate another class of problems, where the classical laplacian is replaced by the N -dimensional m-laplacian.

Finally in chapter 4, we are concerned about singular boundary value problems for second order ordinary differential equations on unbounded domains, which arise, for ex-ample, in hydromechanics, [36], and in nonlinear field theory and cosmology, [30]. The analysis of such problems using our approach was started in [36] and continued here, where we extend the results to a wider class of problems.

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Chapter 1 Introduction

1.1 State of the Art and Applications

In the study of nonlinear phenomena in physics, engineering and other sciences, many mathematical models lead to singular two-point boundary value problems (BVPs) asso-ciated with nonlinear second order ordinary differential equations (ODEs):

x′′ = F (t, x, x′), a < t < b. (1.1) First of all, let us begin with the usual definition of singular two-point BVP:

Definition 1.1 A two-point BVP associated to the second order differential equation (1.1) is singular if one of the following situations occurs

• a and/or b are infinite;

• F is unbounded at some t0 ∈ [a, b];

• F is unbounded at some particular value of x or x′_.

In recent decades, many works have been devoted to the analysis of these BVPs and many different techniques have been used or developed in order to deal with three main questions:

(A) Existence and uniqueness of solutions;

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(C) Computation of solutions.

For the problems studied in this dissertation, whose motivation came initially from real world applications, we focus our attention in questions (B) and (C). We are particu-larly interested in obtaining a rigorous description of the behavior of the solutions in the neighborhood of the singular points. Then, based on this behavior and known existence and uniqueness theorems, we will construct simple efficient numerical algorithms that, as we will see, enable us to compute the solution of the singular boundary value problems accurately.

The motivation for the work presented in the second chapter comes partially from a problem arising in fluid dynamics. The mathematical model which has applications in several problems of engineering, meteorology and oceanography (see, for example [10], [20], [44], [48], [50], [58], [59], [60] and the references therein) is the following

g′′_{(t) = −tg}−1λ(t), 0 ≤ ξ < t < 1, λ > 0 (1.2)

g′_{(ξ) = 0,} _{g(1) = 0,} _(1.3)

and it arises when considering a flat plate with a porous surface that it is moving at a constant speed UW, in the direction parallel to a uniform flow with a constant speed U∞,

where ξ = UW

U∞ is the velocity ratio. For the interested reader, the correct mathematical

formulation of this problem may be found, for example, in [44] and [61]. It is important to remark that from that mathematical formulation, only positive solutions of (1.2)-(1.3) are physically significant. The case λ = 1 corresponds to a Newtonian fluid, the case 0 < λ < 1 describes the power law for pseudoplastic fluids and the case λ > 1 the power law for dilatant fluids.

This problem has received a lot of attention since the middle of last century. One of the first (if not the first) rigorous analysis of this problem was provided by Weyl [54] in 1942, for the case n = 1 and ξ = 0 that corresponds to the classical Blasius problem, where the author proved existence and uniqueness of the solution by formulating an iterative technique. In 1968, Callegari and Friedman [9] proved the existence and uniqueness of the solution of problem (1.2)-(1.3) for λ = 1 and ξ = 0. In 1978, Callegari and Nachman [10] established the uniqueness and analyticity results for λ = 1. Later, in 1980 [44], the same authors proved the existence and uniqueness of the solution in the case 0 < λ < 1

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1. Introduction and ξ = 0.

A great contribution to the study of BVPs for equation (1.2) was given by Taliaferro [51], in 1979, studying the following singular boundary value problem

g′′_{+ φ(t)g}n_{= 0,} _{0 < t < 1} _(1.4)

g(0) = g(1) = 0, (1.5) where n < 0 and φ(t) is positive and continuous for 0 < t < 1. In that paper, Taliaferro used a shooting argument to prove the existence of a unique positive solution and he derived asymptotic estimates for the solution in the neighborhood of singular points.

In 1995, based on these asymptotic estimates, Baxley, [4], constructed a shooting algorithm to compute the solutions of (1.4)-(1.5), remarking that shooting toward singular points, as Taliaferro did, could be useful to prove the existence of a solution, but it was no longer a good idea for its computation, because it leads to computational instability, as concluded earlier by Keller and Shampine, [26], [49]. In order to avoid this problem, it is necessary to use the knowledge of the behavior of the solution in the neighborhood of the singular points, information which is contained in the asymptotic estimates obtained by Taliaferro. That was also the idea followed in [16] and [29], where the authors improved the asymptotic estimates obtained by Taliaferro [51] for problem (1.2)-(1.3) in the case ξ = 0. To be more specific, they determined one-parameter family of solutions that described the behavior of the solution of suitable singular Cauchy problems in the neighborhood of singular points. Based on these families they constructed a shooting algorithm that allowed them to compute the solution of the BVP accurately.

In the first section of the second chapter, whose results are published in [34], we continue the investigation initiated in [16] and [29]. In particular, we will extend the results obtained there to two singular boundary value problems associated to the Emden-Fowler equation

g′′_{(u) = au}σ_gn_{(u) ,} _{0 < u < u}

0, (1.6)

where σ, n < 0, a < 0 and u0 > 0 are real. The first one with boundary conditions:

g′(0) = g (u0) = lim u→u−

0

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and the second one with g (0) = lim u→0+ug ′_{(u) = g (u} 0) = lim u→u− 0 [(u0− u) g′(u)] = 0. (1.8)

It should be remarked that the Emden-Fowler equation (1.6) arises in many other contexts, namely, in astrophysics, nuclear physics and chemical reactions (an extensive bibliography, with 144 entries, may be found in [57]).

In the second section of the second chapter we present the methods and the results published in [32], [35] and [38], where we generalize the analysis of [34], by considering, instead of equation (1.6), the generalized Emden-Fowler equation

(|g′(u)|m−2g′(u))′ = auσgn(u) , 0 < u < u0, (1.9)

where n < 0, a < 0, m > 1, u0 > 0 and σ are real parameters. When m = 2, equation (1.9)

reduces to equation (1.6). The differential operator in the left-hand side of (1.9) is known as the one-dimensional m-laplacian (it reduces to the classical one when m = 2), and has many applications in fluid mechanics and thermodynamics, when describing processes with nonlinear diffusion, see for example [1], [14] and [52].

Recently [21], Houck and Robinson extended the results obtained by Taliaferro in [51] to a more general nonlinear singular BVP:

(|g′_(u)|m−2_g′_(u))′ _{+ φ(u)g}n_{(u) = 0,} _{0 < u < 1,} _(1.10)

g(0) = g(1) = 0, (1.11) where n < 0, m ≥ 2 and φ a positive function in L1

loc(0, 1). They established sufficient

conditions for the existence and uniqueness of solutions of problem (1.10)-(1.11) and, as it was done for the case m = 2 by Taliaferro, they derived asymptotic estimates for the solution and its derivative at the endpoints of the interval. As pointed out in that work, since equation (1.10) may be written as

g′′+ φg

n

(m − 1)|g′_|m−2 = 0,

an additional singularity occurs when the first derivative of g vanishes. In the second section of the second chapter we will improve those representations in the case φ(u) = auσ

and we will obtain similar asymptotic estimates near this new singular point (the point where the solution attains its maximum, as it will be better explained there).

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1. Introduction

Unfortunately, the results of Houck and Robinson do not apply to the case 1 < m < 2. In this case, concerning the questions of existence and uniqueness of solutions we refer to the works of Jiang and Gao, [22], [23], where the authors, based on the method of upper and lower solutions, established sufficient conditions for the existence of a unique positive solution of a more general problem

(|g′_(u)|m−2_g′_(u))′ _{+ f (u, g, g}′_{) = 0,} _{0 < u < 1,} _(1.12)

g(0) = g(1) = 0. (1.13) Based on the same method, Jin, Yin and Wang, [24], established the existence of positive solutions of a related problem for an elliptic equation. Actually, they investigated the existence of positive radial solutions (that is, solutions that depend only on r = |u|) of the Dirichlet problem

div (|∇g|m−2_{∇g) + f(|u|, g) = 0, u ∈ B}

g(u) > 0, _{u ∈ B} g(u) = 0, _{u ∈ ∂B,}

(1.14)

where B = B(0) is the unit open ball centered at the origin in RN_{, N ≥ 1, m > 1. When}

looking for such solutions, problem (1.14) could be reduced to the following BVP for an ODE:

r1−N _rN −1_|g′_|m−2_g′′_{+ f (r, g) = 0,} _{r ∈ (0, 1)}

g(r) > 0, _{r ∈ (0, 1)} g′_{(0) = g(1) = 0.}

(1.15)

In this case, problem (1.15) has been studied intensively in recent years, mainly the questions of existence and uniqueness of solutions, see for example the work of Wong ([55], [56]) and Naito [45] for the case m ≥ 2, among many others.

Following the approach used in [39], for each problem studied in chapter 2, we will also construct upper and lower solutions and show that, in some particular cases, we are able to obtain a closed formula for the exact solution.

While the second chapter of this dissertation is devoted to the analysis of singular BVPs on specified bounded intervals, the third chapter is concerned with singular free boundary problems, that is, problems that are also defined in bounded intervals, but whose

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boundary is an unknown of the problem. More precisely, let us consider the equation y′′(r) +N − 1

r y

′_{(r) = f (y),} _{0 < r < +∞} _(1.16)

with boundary conditions

y′(0) = 0, (1.17) y(M ) = y′_{(M ) = 0, for some M > 0.} _(1.18)

We say that problem (1.16), (1.17), (1.18) is a free boundary problem.

Existence and uniqueness results for the solution of such problems, with f regular, were obtained, for example, by Kapper and Kwong in [25], considering in particular, the case f (y) = yp _{− y}q_, _{0 ≤ p < q ≤ 1, the so called canonical nonlinearity, motivated}

by its application in plasma physics for Tokamac equilibria with magnetic islands. That model in plasma physics was recently proposed by Miller, Faber and White, [43], and it is a particular case of problem (1.16), (1.17), (1.18) with f (y) given by

f (y) = −ayq+ byp, (1.19) a = 5.9, b = 0.2066, p = 0 and q = 1₂.

Motivated by a similar free boundary problem, arising when studying force-free mag-netic fields in a passive medium ([11], [40], [41]), Chen [12], investigated the existence of positive solutions of problem (1.16), (1.17), (1.18), allowing f to be singular at 0. That model corresponds to a particular case of problem (1.16), (1.17), (1.18) with f (y) given by (1.19), a = 1, b = _α1_{, p = 1 − α, q = 1 and 1 < α < 2.}

Many authors have investigated this kind of free boundary problems, but we believe that, for the classes of problems studied here, most of the main results obtained so far are summarized in [19] and the references therein, where Gazzola, Serrin and Tang allowed f to be singular at 0 and where instead of equation (1.16) they studied the equation

(|y′|m−2y′)′+ N − 1 r |y

′_|m−2_y′ _{= f (y),} _(1.20)

where the left hand side represents the radial part of the N -dimensional degenerate m-laplacian, △m, which arises also in the left hand side of (1.15).

All this together constitutes the motivation for chapter 3, where we will describe, in the neighborhood of the singular points, the behavior of the solutions of problem (1.20),

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1. Introduction

(1.17), (1.18) with m > 1, f (y) given by (1.19), a,b > 0 and p < q. We will also adapt, for this kind of problems, the shooting algorithms constructed in chapter 2. Our main results concerning the case m = 2 and f (y) = −ayq_{+ by}p_, _{0 ≤ p < q ≤ 1, a, b > 0}

may be found in [31].

Although it seems natural to construct a shooting algorithm based on the determined one-parameter families of solutions, in this chapter we will also construct a finite difference scheme and compare the numerical results obtained by the two methods. It is important to remark that we are now giving the first steps in the implementation of such finite difference schemes for singular problems and there are many questions that need to be worked out. Still we leave here the idea for such methods that we intend to improve in the future, as for example, the choice of initial approximations and the convergence order of the discretization method.

Finally, in chapter 4 we study singular BVPs on the positive half-axis. An example of such problems, arising in hydromechanics, was studied in [36]:

rN −1_ρ′′ _{= 4r}N −1_λ2_{(ρ + 1)ρ(ρ − ξ), 0 < r < +∞,} _(1.21) lim r→0+ρ ′_{(r) = 0,} _(1.22) lim r→+∞ρ(r) = ξ, (1.23)

where λ and ξ are real positive parameters and N ≥ 2. As explained there, this model describes the formation of microscopical bubbles in a nonhomogeneous fluid like, for ex-ample, vapor inside a liquid. The authors were interested in solutions of the singular BVP (1.21), (1.22), (1.23), which have exactly one zero on R+_{. If such bubble-type solutions}

exist, many important physical properties, such as the gas density inside the bubble, the bubble radius and the surface tension, will depend on them.

In that paper, the authors presented the mathematical formulation of the physical problem, and based on previous works of existence of solution of this kind of problems, [19], the authors gave a correct statement of the limit boundary conditions at singular points (at infinity and at r = 0), computed approximate solutions and analyzed their behavior.

In [30], those results are expanded and some new ones are introduced. Moreover, other applications of problem (1.21), (1.22), (1.23) are presented, namely, in nonlinear

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field theory and cosmology.

Note that if in problem (1.21), (1.22), (1.23) we perform the variable substitution ρ = ξ − y, it becomes y′′_{(r) +}N −1 r y′(r) = 4λ 2_{(y − ξ − 1)y(y − ξ), 0 < r < +∞} _(1.24) y′_{(0) = 0,} _(1.25) lim r→+∞y(r) = 0. (1.26)

By definition (see, for example [19]), a non-negative non-trivial continuously differen-tiable solution of problem (1.24), (1.25), (1.26) is called a radial ground state of ∆y = f (y), with f (y) = 4λ2_{(y − ξ − 1)y(y − ξ). Under the assumption that f is regular, there are}

in the literature a number of well known existence theorems for radial ground states of ∆y = f (y), see [2], [3], [6], [17], among many others. A more general class of problems has been analyzed in [19], where the authors instead of equation (1.16), studied the equation (1.20). In all these papers, sufficient conditions where imposed on f that guarantee the existence of at least one positive solution to the considered problems. A theorem about uniqueness of solution was also presented in [18].

Recently, Bonheure et al [8] established sufficient conditions for the existence of solu-tions of the problem

y′′_{(r) +} N −1

r y′(r) = c(r)f (y), 0 < r < +∞ (1.27)

y′_{(0) = 0,} _(1.28)

lim

r→+∞y(r) = 0, (1.29)

where c(r) is a continuous function in [0, +∞[, there exist real numbers Cmax and Cmin

such that 0 < Cmin ≤ c(r) ≤ Cmax, ∀r ∈ [0, +∞[.

In chapter 4 we present the following developments: First, we generalize the work in [30] and [36] studying problem (1.27), (1.28), (1.29) in the case of nonconstant c(r); Second, we study problem (1.27), (1.28), (1.29) considering a nonlinearity of the form f (y) = y − yq_{, q > 1, a model that describes the pressure distribution in a Van der}

Walls fluid, see [46] and the references therein. Moreover when f (y) = y − y3 _{and c(r) is}

constant, problem (1.27), (1.28), (1.29) is a well known model in field theory. The main results are published in [33] and [37], where we also studied similar problems on bounded domains, that we will not present here.

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1. Introduction

The thesis finishes with the conclusions of the work and a list of open problems that will be subject of future work.

1.2 Mathematical Background

As mentioned before, in the numerical approximation of the solution of singular boundary value problems, it is essential to know how the solution behaves in the neighborhood of the singular points. The approach used in this dissertation leads us to the analysis of singular Cauchy problems (CPs for short) for equation (1.1).

We will not be exhaustive in the study of such problems. We only present some known results from the theory of ordinary differential equations ([13], [42] and [53]), and other results obtained by Konyukhova, that will be often used in this dissertation. Let us then begin with some basic definitions.

Definition 1.2 A function f is holomorphic (or analytic) at a point if it is representable by a convergent power series in the neighborhood of that point. f is holomorphic in a set, if it is holomorphic at every point of the set.

Definition 1.3 A matrix is called holomorphic if every entry of it is a holomorphic func-tion.

Definition 1.4 We say that t0 ∈ (a, b) is an ordinary point of equation (1.1) if F is

holomorphic at t = t0. If not, t0 is a singular point of (1.1).

The simplest kind of singularities of (1.1) are isolated singularities, also called pole-type singularities, that is, points such that the function F is holomorphic in an annular neigh-borhood of them. Here, an annular neighneigh-borhood of a certain point is a neighneigh-borhood with that point itself deleted.

When (1.1) is a linear homogeneous equation, then it can be written in the form x′′(t) + a1(t)x′(t) + a2(t)x(t) = 0. (1.30)

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Definition 1.5 The differential equation (1.30) is said to have a regular singular point at t = 0 if it has a singular point at t = 0 and it may be written in the form

t2x′′(t) + tb1(t)x′(t) + b2(t)x(t) = 0, (1.31)

where b1 and b2 are holomorphic functions at t = 0.

A singular point that is not regular is called irregular.

Remark 1.1 The choice of the singularity at t = 0 is immaterial since the transforma-tions t = t∗_{− t}

0 or t = _t1∗ will shift the singular point to a finite point t = t0 or to infinity,

respectively, without changing the type of the singularity.

Being (1.1) a linear homogeneous equation, it can also be written in the matrix form X′ = A(t)X, (1.32) where A(t) is a square matrix of order two. If A(t) has a pole-type singularity at t = 0, we can rewrite (1.32) in the form

trX′ = B(t)X, (1.33) where r > 0 is the minimal integer value of the exponent of t for which B(t) is holomorphic at t = 0.

Analogously, we have the following

Definition 1.6 If r = 1 in system (1.33), the point t = 0 is called a regular singular point of system (1.32). If r > 1 the point t = 0 is an irregular singular point of (1.32).

Before we proceed let us make a simple but useful remark. Suppose that (1.30) has a regular singular point at t = 0. Then (1.30) may be written in the form (1.31). Performing the change of variables x1 = x, x2 = tx′, equation (1.31) may be rewritten in the matrix

form (1.33) with r = 1, and

B(t) =   0 1 −b2(t) 1 − b1(t)   .

It can be easily seen that the characteristic equation of B(0) is

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1. Introduction

Definition 1.7 Equation (1.34) is called indicial equation of equation (1.31) at t = 0. The roots of the indicial equation (1.34) are called characteristics exponents.

The above definitions can be extended to the case where (1.1) is nonlinear:

Definition 1.8 Consider the nonlinear second order ordinary differential equation repre-sented in the matrix form by

trX′ = f (t, X), (1.35) where X(t) and f (t, X) are 2-dimensional column vectors, and r > 0 is the minimal integer value of the exponent of t for which f (t, X) is holomorphic at t = 0, X0 = X(0).

If r = 1 in equation (1.35), the point t = 0 is called a regular singular point. If r > 1 the point t = 0 is an irregular singular point of (1.35).

Let r = 1 in (1.35). Exhibiting its linear terms, system (1.35) may be written in the following form

tX′ = A(t)X + ˜f (t, X), (1.36) where ˜f (t, X) contains only nonlinear terms in X. Its characteristic exponents at t = 0 are obtained considering only the linear part of (1.36) and using, as in the linear case, (1.34).

Analogous definitions hold if instead of equation (1.1) we have an ordinary differential equation of order n, see for example [53], but we will restrict ourselves to the case n = 2, since equations of different order will not be studied here.

Given the basic definitions, we can now enunciate the main results that we will use in the next three chapters.

Consider the following second order nonlinear differential equation

t2x′′(t) + tb1(t)x′(t) + b2(t)x(t) = f (t, x, tx′) + g(t), 0 < t ≤ t0, (1.37)

where b1(t) and b2(t) are holomorphic functions at t = 0. According to our previous

definitions, equation (1.37) has a regular singular point at t = 0.

We will always assume that f (t, x, y) contains only nonlinear terms in x and y, is an holomorphic function on the set of variables (t, x, y) ∈[0,t0]×{(x, y) ∈ R2 : max{|x|, |y|} ≤

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a}, satisfying f(t, 0, 0) = 0, for all the considered values of t and g(t) is continuous on [0, t0].

Suppose that we want to find the solutions of (1.37), in the neighborhood of t = 0, that satisfy the conditions

x(0) = 0, lim

t→0tx

′_{(t) = 0.} (1.38)

Let us first consider that g(t) ≡ 0 (homogeneous case), and that the characteristic exponents of (1.37) have opposite signs.

In this case, performing the variable substitution

t = e−u, (1.39)

(1.37) becomes

¨˜x + (1 − b1(e−u)) ˙˜x + b2(e−u)˜x = ˜f (u, ˜x, ˙˜x), u ≥ U0, (1.40)

and the conditions (1.38) are replaced by lim

u→∞x(u) = 0,˜ u→∞lim ˙˜x(u) = 0. (1.41)

Setting

x1 = ˜x, x2 = ˙˜x, (1.42)

we can rewrite (1.40), (1.41) in the form   ˙x1 ˙x2   =   0 1 −b2(e−u) b1(e−u) − 1     x1 x2   +   0 ˜ f (u, x1, x2)   (1.43) lim

u→∞x1(u) = 0, u→∞lim x2(u) = 0. (1.44)

The case where (1.43) is asymptotically autonomous was studied in detail in [27]. If this is the case, then lim

u→∞b1(e

−u_{) = b} 1, lim

u→∞b2(e

−u_{) = b}

2, where b1 and b2 are constants and

the function lim

u→∞

˜

f (u, x1, x2) = ˜˜f (x1, x2) does not depend explicitly on u, and therefore,

when u → ∞ the system (1.43) reduces to the autonomous system:   ˙x1 ˙x2   =   0 1 −b2 b1− 1     x1 x2   +   0 ˜˜ f (x1, x2)   . (1.45)

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1. Introduction

Note that the eigenvalues of the matrix in (1.45) are precisely the symmetric of the characteristic exponents of (1.37) at t = 0 (see (1.34)).

Accordingly to this definition of autonomous and asymptotically autonomous system, we have the following definition of autonomous and asymptotically autonomous equation: Definition 1.9 We say that equation (1.37) with g(t) ≡ 0 is an autonomous equation if b1(t) and b2(t) are constant functions and the function f on the right hand-side of

(1.37) does not depend explicitly on t. If in (1.37), lim

t→0b1(t) and limt→0b2(t) are constant

functions and if lim

t→0f (t, x, tx

′_{) does not depend explicitly on t, we say that (1.37) is an}

asymptotically autonomous equation.

The main results obtained in [27], concerning the Cauchy problem (1.43), (1.44), where (1.43) is an asymptotically autonomous system, may be summarized in the following theorem:

Theorem 1.1 Let Reλ1 < 0, Reλ2 > 0, where λ1 and λ2 are the characteristic roots of

the matrix on the right hand-side of (1.45). If the function ˜f (u, x1, x2) defined on the

right hand-side of (1.43) contains no linear terms in x1 and x2 and it is an holomorphic

function on the set of variables (u, x1, x2) ∈ [U0, ∞[×{(x1, x2) ∈ R2 : max{|x1|, |x2|} ≤ a},

satisfying f (u, 0, 0) = 0, for all the considered values of u, then (1.43) has a one-parameter family of solutions which satisfy (1.44) and that may be represented in the form of the convergent series X(u, c) = α1(u)ceλ1u+ ∞ X k=2 αk(u)ckekλ1u, u ≥ U0, (1.46) where X =   x1 x2 

, c is an arbitrary constant and the coefficients αk, k = 1, 2, . . . (which

do not depend on c) are continuous bounded vector functions on [U0, ∞[. Moreover, when

u → ∞, these coefficients satisfy

αk(u) ∼ ∞ X m=0 αkm um , k ≥ 1, with lim u→∞α1(u) =   1 0   .

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Returning to the original variables, it follows that, if (1.37) is an asymptotically au-tonomous and homogeneous equation, then problem (1.37), (1.38) has a one-parameter family of solutions that can be represented by the convergent series

x(t, c) = η1(t)ct−λ1 + ∞

X

k=2

ηk(t)ckt−kλ1, (1.47)

where the coefficients ηk(t), k = 1, 2, . . ., are continuous bounded functions of t, and in

particular lim t→0η1(t) = 1, lim t→0η ′ 1(t) = 0.

The autonomous case was firstly studied by Lyapunov, [42], who considered systems of n equations and introduced the series representations (1.46). That is the reason why these expansions are often referred in the literature as Lyapunov series. In this case, the coefficients αk, k = 1, 2, . . ., in (1.46) are constants. That is, if (1.37) is an autonomous

and homogeneous equation, there is, in the neighborhood of t = 0, a one-parameter family of solutions that satisfy the initial conditions (1.38) and that may be represented by the convergent Lyapunov series

x(t, c) = ct−λ1 ₊

∞

X

k=2

ηkckt−kλ1, (1.48)

where c is the parameter, the coefficients ηk, k = 1, 2, . . . are constants and µ1 = −λ1 is

the only characteristic exponent of (1.37) with positive real part.

Finally, if (1.37) is non-homogeneous and if xp(t) is a particular solution, we can easily

see that the transformation y = x−xpwill transform (1.37) into an homogeneous equation

whose linear part coincides with the linear part of (1.37).

Let us now consider the case where the characteristic exponents of (1.37) have both negative real parts, and that b1(t) and b2(t) are constant functions, say b1(t) ≡ b1 and

b2(t) ≡ b2 .

Performing, as before, the variable substitutions (1.39) and (1.42), problem (1.37), (1.38) may be rewritten as   ˙x1 ˙x2   =   0 1 −b2 b1− 1     x1 x2   +   0 ˜ f (u, x1, x2)   +   0 ˜ g(u)   (1.49) lim

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1. Introduction

where now, the eigenvalues of the matrix in (1.49) have positive real parts.

Under some conditions on ˜f (u, x1, x2) and ˜g(u), the existence of a unique solution of

the Cauchy problem (1.49), (1.50) can be assured by using the results in [7], where the authors studied systems of N -order nonlinear first order equations of the form

X′ = urAX + urf (u, X) + urg(u), U0 ≤ u < ∞, (1.51)

assuming that r is a non-negative integer, A is a constant matrix whose eigenvalues have positive real parts, f (u, X) is defined and continuous for small |X| = PNj=1|Xj| and

u ≥ U0, g(u) is defined and continuous for u ≥ U0 and lim

u→∞g(u) = 0.

In the particular case of homogeneous equation (1.51), the authors proved the following result:

Lemma 1.1 If in (1.51), g(u) ≡ 0, f(u, 0) = 0, and if for any ε > 0, there exists δε, Tε,

such that

|f(u, X)| ≤ ε|X|,

for |X| ≤ δε and all u ≥ Tε, then (1.51) has no other solution that approaches zero as

u → ∞ apart from X(u) ≡ 0.

Note that system (1.49) is a particular case of (1.51) with r = 0, allowing us to conclude the following

Corollary 1.1 If in (1.37)

• b1(t) ≡ b1, b2(t) ≡ b2, where b1 and b2 are constants,

• f(t, x, y) is defined and continuous for small |x| and |y| and t ≤ t0 and there exists

δε, τε, such that

|f(t, X)| ≤ ε|X|, for |X| ≤ δε and all t ≤ τε, where X =

  x y  , • f(t, 0, 0) = 0, • g(t) ≡ 0,

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and if its characteristic exponents have negative real parts, then problem (1.37), (1.38) has no other solution than x(t) ≡ 0.

The case where one of the characteristic exponents has real part equal to zero is a special one. This case was studied in [28], where Konyukhova investigated singular Cauchy problems for systems of the form (1.35), assuming that r ≥ 0 is an integer. In that paper, the author distinguished the case r = 1 (the case of a regular singular point, according to our previous definitions) from the case r > 1 (in this case we will have an irregular singular point). As it is customary, the author has shifted the singularity to the point at infinity, and considered singular Cauchy problems of the form

t−rx′ = A0+ L X j=1 Aj tj ! x + f (t, x) + g(t), _{T ≤ t < ∞} (1.52) lim t→∞x(t) = 0, (1.53)

where r ≥ 0 is an integer, T is a fairly large positive constant, L is an integer, 0 ≤ L ≤ r + 1, where L = 0 means that there is no sum, A0, Aj, j = 1, 2, . . . , L are constant

square matrices of order N , the vector function is continuous over the set of variables {x ∈ RN _{: |x| ≤ a} × [T, ∞], where |x| = max}

j=1,...,N|xj|, f(t, 0) = 0, g(t) is an N-column

vector continuous in [T, ∞), lim

t→∞g(t) = 0 and the real parts of the eigenvalues of A0 are

non negative (Reλ(A0) ≥ 0, for short).

With the purpose of obtaining existence and uniqueness results, Konyukhova has con-sidered the two following sets of conditions:

Assumptions I Suppose that in system (1.52) L = r + 1 and Reλ(A0) ≥ 0 so that

the eigenvalues λ(A0) lying on the imaginary axes are simple; a continuous positive

function m(t) exists such that Z ∞

T

m(t)tr_{dt < ∞,} (1.54) |f(t, x) − f(t, ex)| ≤ m(t)|x − ex|, (1.55) when t ≥ T , x, ex ∈ {x ∈ RN _{: |x| ≤ a}; the N-column g(t) satisfies the condition}

Z _∞

T |g(t)|t r

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1. Introduction

Assumptions II Suppose that in system (1.52) L = 0, Reλ(A0) ≥ 0 and to the purely

imaginary eigenvalues λ(A0) there correspond Jordan cells of order not greater than

χ, χ ≥ 1; for f(t, x) and g(t) assumptions analogous to Assumptions I hold, in which (1.54) and (1.56) are replaced by the conditions

Z ∞ T m(t)tχ(r+1)−1_{dt < ∞,} Z ∞ T |g(t)|t χ(r+1)−1_{dt < ∞;} _(1.57)

and proved the following

Theorem 1.2 Suppose that Assumptions I together with r ≥ 0 an integer, or Assump-tions II with r > −1 not necessarily integer, hold for system (1.52). Then for sufficiently large T a solution of problem (1.52), (1.53) exists and is unique.

Moreover, under more restrictive conditions on f (t, x) and g(t) she derived an asymptotic expansion for the solution of the considered problem:

Theorem 1.3 Suppose that in system (1.52) g(t) = eg(ζ), f(t, x) = ef (ζ, x), where ζ = 1_t. Define the quantity Π by: Π = ω(r + 1) + 1, where ω = 1 when Assumptions I are satisfied and ω = χ in the case of Assumptions II. Suppose that eg(ζ) has Q, Q = Π + n continuous derivatives in [0, ζ0], and the function eg(ζ)ζ−Π has a finite limit as ζ → 0; ef (ζ, x) has

continuous partial derivatives up to the Q-order inclusive with respect to the variables x1, . . . , xn in the region {|x|s ≤ a, 0 ≤ ζ ≤ ζ0}, where |x|s = (x, x)

1

2, (x, y) = PN

j=1xjyj,

e

f (ζ, 0) = 0 and the limit lim

ζ→0 ∂f ∂x

ζ−Π _{exists uniformly with respect to x when |x|} s ≤ a.

Then, for sufficiently large T there is a unique solution θ(t) of problem (1.52), (1.53) that may be represented by

θ(t) =Pn_m=η θ_t(m)m + o(t−n), t → ∞, (1.58)

where η = Π if detA0 6= 0 and η = 1 otherwise.

Moreover, if Q = ∞, that is, if eg(ζ) and ef (ζ, x) have derivatives of all orders in the region indicated, then for the solution of problem (1.52), (1.53) the following asymptotic representation holds for large t:

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Remark 1.2 In this thesis, theorem 1.3 will be applied only to systems of two equations, with L = 0. Moreover, the considered matrix A0 has two real distinct eigenvalues: one

is positive and the other one equals to zero. Therefore, Assumptions II must be satisfied with χ = 1.

Corollary 1.2 Suppose r = −1, L = 0, Reλ(A0) ≥ 0 in system (1.52). Suppose that the

conditions of theorem 1.3 hold for g(t) and f (t, x) when Π = 1. Then the assertions of theorem 1.3 remain true.

Under the hypothesis of corollary 1.2, and performing the variable substitution et = 1_t, we obtain from problem (1.52), (1.53) the following singular Cauchy problem

tx′ = A0x + f (t, x) + g(t), 0 < t ≤ t0 (1.60)

lim

t→0x(t) = 0, (1.61)

where A0 is a square constant matrix of order N , Reλ(A0) ≤ 0, the vector function

f (t, x) is continuous over the set of variables (t, x) ∈ [0, t0] × {x ∈ RN : |x| ≤ a}, where

|x| = maxj=1,2,...N|xj|, f(t, 0) = 0, and g(t) is an N-column continuous function in [0, t0]

such that g(0) = 0.

Then Konyukhova obtained the analog of theorem 1.3 for problem (1.60), (1.61): Theorem 1.4 Suppose Reλ(A0) ≤ 0 in system (1.60); g(t) has n + 1 continuous

deriv-atives in [0, t0] and g(0) = 0; suppose f (t, x) has continuous partial derivatives up to

the (n + 1)-th order inclusive, with respect to the variables x1, . . . , xN in the region

{|x|s ≤ a, 0 ≤ t ≤ t0}, f(t, 0) = 0, and a finite limit lim t→0

∂f ∂xt−1

exists uniformly with respect to x when |x|s ≤ a.

Then, for sufficiently small t0 a unique solution θ(t) of problem (1.60), (1.61) exists

and θ(t) = n X m=1 θ(m)tm+ o(tn), _{t → 0}

where the coefficients θ(m)_{, m ≥ 1 may be determined from (1.60) by a formal substitution} of all the expansions.

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1. Introduction

Theorem 1.5 Suppose that the following assumptions hold for system (1.60): Reλ(A0) ≤

0, g(t) and f (t, x) can be represented by the convergent series g(t) = ∞ X m=1 g(m)tm, _{t ≤ t}0, (1.62) f (t, x) = X |l|≥1,m≥0,|l|+m≥2 f_l(m)xltm_{, x ∈ {x ∈ R}N _{: |x| ≤ a} × [0, t}0], t ≤ t0, (1.63)

where xl _{denotes the product x}l1

1...x lN

N where lj, j = 1, . . . , N are non-negative integers.

Then the following assertions hold:

1. For sufficiently small t0 a solution θ(t) of problem (1.60), (1.61) exists, represented

by the convergent series

θ(t) =

∞

X

m=1

θ(m)tm, _{t ≤ t}0, (1.64)

where the coefficients θ(m) _{are found from (1.60) by a formal substitution of all the}

expansions, and this is the unique solution of problem (1.60), (1.61) in the class of analytical functions;

2. Suppose 0 < ν ≤ 1; then (1.64) is the unique solution of (1.60) which satisfies the condition

|θ(t)| = O(tν), _{t → 0;}

3. Suppose f_l(0) _{= 0 in (1.63) for all l, |l| ≥ 1; then (1.64) is the unique solution of} problem (1.60), (1.61).

Remark 1.3 If in (1.60), (1.61), f (t, x) = ˜f (τ, x) and g(t) = ˜g(τ ), τ = tη_{, where η is}

any positive real number, we easily see that we can allow non-integer powers of t in (1.62), (1.63), that is, we may replace these expansions by the following ones

g(t)=P∞_m=1g(m)_tmη_, _{t ≤ t} 0,

f (t, x) =P_{|l|≥1,m≥0,|l|+m≥2}f_l(m)xl_tmη_{, x ∈ {x ∈ R}N _{: |x| ≤ a} × [0, t}

0], t ≤ t0,

and the existence of a unique solution of problem (1.60), (1.61) is still guaranteed. In fact, if this is the case, performing the variable substitution τ = tη_{, (1.60), (1.61)}

rewrites τ ˙x = ˜A0x + 1 ηf (τ, x) +˜ 1 ηg(τ ),˜ 0 < τ ≤ τ0 (1.65) lim τ →0x(τ ) = 0, (1.66)

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where ˜A0 = 1_ηA0.

Since for ˜f (τ, x) and ˜g(τ ) expansions of the form (1.62), (1.63) hold, and since the eigenvalues of the matrix ˜A0 have the same signs as the eigenvalues of A0, we can apply

theorem 1.5 to conclude that for sufficiently small τ , problem (1.65), (1.66) has a unique solution that can be represented by a convergent series of the form (1.64).

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Chapter 2 Singular Boundary Value Problems

Over Bounded Domains

2.1 Singular boundary value problems for an

Emden-Fowler equation

In this section, we are concerned about singular boundary value problems for the Emden-Fowler equation

g′′_{(u) = au}σ_gn_{(u) ,} _{0 < u < u}

0, (2.1)

where we assume that σ, n < 0, a < 0 and u0 > 0 are real. We seek positive solutions of

this equation that satisfy the two following sets of boundary conditions: g′(0) = g (u0) = lim u→u− 0 [(u0− u) g′(u)] = 0, (2.2) g (0) = lim u→0+ug ′_{(u) = g (u} 0) = lim u→u− 0 [(u0− u) g′(u)] = 0. (2.3)

We shall call (2.1), (2.2) the first singular boundary value problem and (2.1), (2.3) the second singular boundary value problem.

Because n is negative, the second boundary value problem is singular at both endpoints of the interval. The first boundary value problem, which is singular, for the same reason at u = u0, will also be singular at u = 0 (with respect to the independent variable) if σ is

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2.1.1 Existence and uniqueness of solution

As mentioned in the introduction, a great step in the investigation of these BVPs is due to Taliaferro, [51], where the author stated and proved the following existence and uniqueness result:

Theorem 2.1 The boundary value problem

g′′+ φ(t)gn = 0, 0 < t < 1 (2.4) g(0) = g(1) = 0, (2.5) where n < 0 and φ(t) is positive and continuous for 0 < t < 1, has a positive twice continuously differentiable solution iff

Z 1

0 t(1 − t)φ(t)dt < ∞.

In this case that solution is unique. Furthermore, lim

t→0g ′_(t) _lim t→1g ′_(t) _{is finite iff} Z 1 2 0 φ(t)tn_{dt < ∞} Z 1 1 2 φ(t)(1 − t)ndt < ∞ ! .

If in problem (2.1), (2.3) we perform the variable substitution t = _uu

0, it becomes

a particular case of problem (2.4), (2.5), with φ(t) = −auσ+2

0 tσ. Therefore, the second

boundary value problem (2.1), (2.3) has a solution iff the integral R₀1_{t(t − 1)au}σ+2₀ tσ_{dt is}

convergent, what happens when σ > −2, as the reader may easily conclude. According to Theorem 2.1 this solution is unique and will have finite slope at u = 0 iffR 12

0 au σ+2

0 tσ+ndt <

∞, what is equivalent to n > −1−σ, and finite slope at u = u0iff

R1

1 2 au

σ+2

0 tσ(1−t)ndt < ∞

or, equivalently, n > −1. We have just proved the following

Corollary 2.1 Assume that in (2.1) n < 0 and a < 0. Then the BVP problem (2.1), (2.3) has a unique positive twice continuously differentiable solution iff σ > −2. Moreover, that solution is unique and will have finite slope at u = u0 iff n > −1, and finite slope at

u = 0 iff n > −1 − σ.

Later, in 1991, Baxley [5] extended these results to a wider class of BVPs g′′_{(u) + f}

1(u, g′) + f2(u, g) = 0, α < u < β,

a0g(α) − a1g′(α) = A,

b0g(β) − b1g′(β) = B,

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2. Singular Boundary Value Problems Over Bounded Domains assuming

a0, a1, b0, b1, A, B ≥ 0 and a0b0+ a1b0+ a0b1 > 0. (2.7)

In order to get a positive solution of problem (2.6) which was sufficiently smooth at a (possible singular) endpoint of [α, β], the author needed to impose certain integrability conditions on f2(u, g) near that endpoint:

Definition 2.1 Let m be the midpoint of [α, β], and consider the several conditions: (i) R_αmf2(u, θ(u − α)) du < ∞ for θ > 0,

(ii) R_αmf2(u, g)du < ∞ for g > 0,

(iii) R_αm_{(u − α)f}2(u, g)du < ∞ for g > 0,

(iv) R_mβf2(u, θ(β − u)) du < ∞ for θ > 0,

(v) R_mβf2(u, g)du < ∞ for g > 0,

(vi) R_mβ_{(β − u)f}2(u, g)du < ∞ for g > 0,

We say that f2(u, g) satisfies the strong integrability condition at u = α if f2(u, g) satisfies

(ii) in case A + a1 > 0, and f2(u, g) satisfies (i) in case A + a1 = 0. Analogously,

f2(u, g) satisfies the strong integrability condition at u = β if f2(u, g) satisfies (v) in

case B + b1 > 0, and f2(u, g) satisfies (iv) in case B + b1 = 0. We also require that

f2 : (α, β] × (0, ∞) → (0, ∞) is continuous if B = 0 and f1(u, z) fails to satisfy the

condition: There exists δ > 0 such that

f1(u, z) = O(|z|) as z → 0−, uniformly for u ∈ [b − δ, δ]. (2.8)

We say that f2(u, g) satisfies the weak integrability condition at u = β if f2(u, g) satisfies

(vi). Similarly, f2(u, g) satisfies the weak integrability condition at u = α if f2(u, g)

satisfies (iii) when f1(u, z) = O(z) as z → ∞; otherwise we must strengthen this condition

to (ii).

Then, Baxley stated and proved the following existence result: Theorem 2.2 Suppose that f1 and f2 satisfy the following conditions

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(H1) f2 : (α, β) × (0, ∞) → (0, ∞) is continuous;

(H2) f2(u, g) is nonincreasing in g for each fixed u ∈ (α, β);

(H3) f1 : [α, β] × (−∞, ∞) → (−∞, ∞) is continuous;

(H4) zf1(u, z) ≥ 0 for all (u, z) ∈ [α, β] × (−∞, ∞);

If, in (2.6), b0 > 0, put I = [α, β); if b0 = 0, put I = [α, β].

(H5) f1(u, z) satisfies an uniform Lipschitz condition in z on each compct subset S ⊂

I × (−∞, ∞). That is, given such a set S, there exists K > 0 such that |f1(u, z2) − f1(u, z1)| ≤ K|z2− z1|,

whenever (u, z2), (u, z1) ∈ S;

(H6) If a1 = 0 and b0 > 0, then f1(u, z) = O(z2), as z → ∞, uniformly for u ∈ [α, β];

(H7) If b0 = 0, then f1(u, z) = O(z log z), as z → ∞, uniformly for u ∈ [a, b].

At each point of [α, β], suppose either f2(u, g) satisfies the strong integrability

condi-tion, or that the boundary condition at that endpoint is a Dirichlet condition and f2(u, g)

satisfies the weak integrability condition. Let J denote the interval obtained by removing from [α, β] any endpoint at which f2(u, g) fails to satisfy the strong integrability condition.

Then the BVP (2.6) has at least one positive solution φ ∈ C2_{(α, β) ∩ C}1_{(J) ∩ C[α, β].}

When b0 > 0 which is the case we are interested in, Baxley proved the uniqueness

result:

Theorem 2.3 Suppose that f2 satisfies (H1), (H2) and f1 satisfies (H3) and (H6) from

theorem 2.2, and suppose that f1(u, z) is nondecreasing in z for each u ∈ [α, β]. If b0 > 0,

then problem (2.6) has at most one positive solution.

Our first BVP is a particular case of problem (2.6) with f1(u, g′) ≡ 0, f2(u, g) =

−auσ_gn_{, α = 0, β = u}

0, a0 = 0, a1 = 1, A = 0, b0 = 1, b1 = 0 and B = 0. Theorems 2.2

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2. Singular Boundary Value Problems Over Bounded Domains

Corollary 2.2 Assume that in (2.1) n < 0, a < 0 and σ > −1. Then the BVP problem (2.1), (2.2) has a unique positive solution φ ∈ C[0, u0] ∩ C1[0, u0] ∩ C2(0, u0) if n > −1

or φ ∈ C[0, u0] ∩ C1[0, u0) ∩ C2(0, u0) if n ≤ −1.

Proof. It is clear that f2(u, g) = −auσgn, with a, n < 0 satisfies (H1), and since ∂f_∂g2 =

−anuσ_gn−1_{< 0 for each fixed u ∈ (0, u}

0), f2 satisfies (H2). Obviously, f1(u, g′) ≡ 0 satisfy

all the conditions imposed on function f1.

With respect to the integrability conditions, it should be remarked first that at u = 0, we have a Neumann boundary condition (that is, in the boundary condition, the derivative is specified at a particular location of the independent variable), and at u = u0 we have a

Dirichlet condition (in this case is the value of the dependent variable at a certain location of the independent variable that it is specified).

According to theorem 2.2, in order to exist a positive solution of problem (2.1), (2.2), f2 must satisfy the strong integrability condition at u = 0 and the weak integrability

condition at u = u0, that is, the integrals

Rm

0 auσgndu and

Ru0

m (u0 − u)auσgndu,

respec-tively, must be convergent, where m is the midpoint of [0, u0]. We easily verify that the

first integral will be convergent if σ > −1 and that the second one is convergent for all σ, n ∈ R.

At u = u0 the strong integrability condition is satisfied iff

Ru0

m (u0− u)

n_auσ_θn_{du < ∞,}

for θ > 0, what happens when n > −1. Therefore, the interval J of theorem 2.2 is the interval [0, u0] if n > −1 and [0, u0) if n ≤ −1.

2.1.2 Behavior of the solution in the neighborhood of singular

points

Taliaferro, [51], also derived the asymptotic behavior of the solution of (2.4), (2.5) in the neighborhood of the singular points. Their main results for the behavior of the solution in the neighborhood of the singular point t = 0 are resumed in theorems 2.4 and 2.5 below.

Theorem 2.4 Suppose R 12

0 φ(ξ)ξ

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there exists A > 0 such that g(t) = At − An(1 + o(1)) Z t 0 (t − ξ)φ(ξ)ξ n_dξ, as t → 0. Theorem 2.5 Suppose R 12 0 φ(ξ)ξ

n_{dξ = ∞ and g(t) is the solution of (2.4), (2.5). Let}

f (t) =R 12

t ψ(τ )τndτ

1 1−n

where ψ(t) is positive and continuously differentiable on 0, 1₂, and φ(t) ∼ ψ(t) as t → 0. If lim t→0t f′′ (t) f′ (t) = R where R > −2 then g(t) ∼ 1 − n 2 + R 1 1−n tf (t), as t → 0.

To illustrate these results, Taliaferro considered the case φ(t) ∼ tσ _{as t → 0, σ > −2, and}

concluded that the solution g(t) of problem (2.4), (2.5) satisfied g(t) = At − (1 + o(1))(σ+n+1)(σ+n+2)Antσ+n+2 , if σ > −1 − n, g(t) ∼(σ+2)(σ+n+1)−(1−n)2 1 1−n tσ+21−n, if −2 < σ < −1 − n, g(t) ∼ (1 − n)1−n1 t (− log t) 1 1−n, if σ = −1 − n, (2.9) as t → 0.

In this section, we will improve representations (2.9), but first, we will obtain the behavior of the solution of the two boundary value problems (2.1),(2.2) and (2.1),(2.3), in the neighborhood of the singular point u = u0. In order to do that, we will follow the

approach used in [16] and we consider the singular Cauchy problem:

g′′(u) = auσgn(u) , 0 < u < u0, (2.10)

g (u0) = lim u→u−

0

[(u0− u) g′(u)] = 0, (2.11)

where we assume that n < 0, a < 0, u0 > 0 and σ > −2 are real numbers. One of our

main results in this section is the following

Proposition 2.1 The singular Cauchy problem (2.10), (2.11) has, in the neighborhood of the singular point u = u0, a one-parameter family of solutions that may be represented

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2. Singular Boundary Value Problems Over Bounded Domains by g(u, b) =                g1(u, b), n < −1 and n 6= −3; g2(u, b), n = −3; g3(u, b), n > −1; g4(u, b), n = −1;

where b is the parameter and g1(u, b) = auσ 0(1−n)2 2(1+n) 1 1−n (u0− u) 2 1−n h 1 − _{(1−n)(3+n)u}(1+n)σ 0 (u0− u) + b (u0 − u) −2(1+n) 1−n + O((u 0− u)1+µ) i , (2.12) where µ = minn1,−2(1+n)_1−n o, g2(u, b) = _auσ 0(1−n)2 2(1+n) 1 4 (u0− u) 1 2 h 1 + b (u0− u) h 1 + −288b2u20−48bu0σ+11σ2 576bu2 0 (u0− u) + · · · ] + (u0− u) ln u0−u u0 h σ 8u0 − σ(12bu0+σ) 96u2 0 (u0− u) + · · · i + (u0− u)2ln2 u0−u u0 h −128uσ22 0 + · · · i + · · ·i, (2.13) g3(u, b) = b(2+n)(n+1)_auσ 0 1 n (u0− u) + b (u0− u)2+n 1 + y0,1(b) (u0− u)1+n+ o (u0− u)1+n , (2.14) where y0,1 is given by y0,1(b) = bn (2 + n) 2 (2n + 3) b(2 + n) (n + 1) auσ 0 −1n , (2.15) and g4(u, b) =√−2auσ0(u0− u) r − lnu0−u u0 1 + ln − ln u0−u u0 4 − b 1 ln u0−u u0 + −ln 2 − ln u0−u u0 32 + 1 8 + b 4 ln_{− ln}u0−u u0 − 3 8 + b 2 + b2 2 o 1 ln2 u0−u u0 + · · · . (2.16)

Proof. Let us consider first the case n < −1. Note that taking into account corollary 2.1, in this case the solutions will have infinite slope at u = u0.

In the neighborhood of u = u0, we shall look for a solution of the singular Cauchy

problem (2.10), (2.11) in the form:

g (u) = C (u0− u)k[1 + o(1)], (2.17)

g′_{(u) = −Ck (u}

0− u)k−1[1 + o(1)],

g′′_{(u) = Ck (k − 1) (u}

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From (2.10), (2.11) we have lim u→u− 0 g′′_{(u) g}−n_(u)_{= au}σ 0. (2.19)

Substituting (2.17) and (2.18) in (2.19) we obtain lim u→u− 0 C1−n_{k (k − 1) (u}0 − u)k−2−kn = auσ₀, which implies that k = 2

1−n > 0, k (k − 1) = 2(1+n) (1−n)2 < 0 and C = ( auσ 0(1−n)2 2(1+n) ) 1 1−n. In order

to improve the representation (2.17), in (2.10), (2.11) we perform the variable substitution g (u) = C (u0− u)k[1 + y (u)],

and obtain the Cauchy problem in y (u0− u)2y′′−_1−n4 (u0− u) y′+ 2(1+n)_(1−n)2 h 1 + y − (uu0) σ_{(1 + y)}ni_{= 0, 0 < u < u} 0,(2.20) y (u0) = lim u→u− 0 [(u0− u) y′(u)] = 0, (2.21)

where u = u0 is a regular singular point. Writing out the leading linear homogeneous

terms of (2.20), for solutions that satisfy (2.21), we obtain the equation (u0− u)2y′′− 4 1 − n(u0− u) y ′₊2 (1 + n) 1 − n y = 0, u → u − 0 (2.22)

with characteristic exponents λ1 = −1 < 0 and λ2 = −2(1+n)_1−n > 0. Since only one of the

characteristic exponents is positive, for any value of n < −1 the problem (2.20), (2.21) has a one-parameter family of solutions.

First, we shall consider n 6= −3. In this case (2.20), (2.21) has a particular solution, yp(u), holomorphic at u = u0 and if u 6= u0,

yp(u) = +∞

X

l=1

yl(u0− u)l, |u0− u| ≤ δ, δ > 0, (2.23)

where the coefficients yl may be determined substituting yp(u) in (2.20). For example, for

l = 1, we get, y1 = −_{(1−n)(3+n)u}(1+n)σ ₀.

Changing to the variable ˜_{y = y − y}p, problem (2.20), (2.21) may be rewritten as

(u0− u)2y˜′′− _1−n4 (u0− u) ˜y′+2(1+n)_(1−n)y = f (u, ˜˜ y, ˜y′), 0 < u < u0, (2.24)

y (u0) = lim u→u−

0

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2. Singular Boundary Value Problems Over Bounded Domains where f (u, ˜y, ˜y′) = 2(1 + n) (1 − n)2 (1 + ˜y + yp)n− n˜y − 1 − u u0 σ (1 + ˜y + yp)n − (1 + yp)n+ 1 − u u0 σ (1 + yp)n] ,

contains no linear terms in ˜y and satisfies f (u, 0, 0) = 0.

According to definition 1.9, (2.24) is an asymptotically autonomous equation. It fol-lows from theorem 1.1 that problem (2.24), (2.25) will have a one-parameter family of solutions represented by the convergent series

˜ y(u, b) = α1(u)b(u0− u)λ2 + ∞ X l=2 αl(u)blyl(u0 − u)lλ2, (2.26)

where the coefficients αl(u), l = 1, 2, . . ., depending on u but not on b, may be determined

substituting (2.26) in (2.24). For l = 1 we obtain the following CP on α1

(u0− u)2α′′1+ _1−n4n (u0− u)α′1+ 2n(n+1) (1−n)2 α1 = 2n(n+1)_(1−n)2 u u0 σ , lim u→u0 α1(u) = 1, lim u→u0 α′ 1(u) = 0. (2.27)

Note that u = u0 is a regular singular point of the second order differential equation in

(2.27), whose characteristic exponents are 1+n_1−n < 0 and _1−n2n < 0. Performing the variables substitutions t = u0− u and ˜α1 = α1− 1, we easily see that we are in conditions to use

theorem 1.5 to conclude that problem (2.27) has a unique solution that may be represented by α1(u) = ∞ X j=1 aj(u0− u)j,

whose constant coefficients aj, j = 1, 2, . . ., may be determined by formal substitution in

(2.27), resulting for j = 1, a1 = _2(1−n)u(n+1)σ₀, for instance.

For the remaining coefficients αj, j = 2, 3, . . ., analogous representations hold,

pro-ceeding in the same way.

Returning to the original variables we conclude that the one-parameter family of so-lutions of the Cauchy problem (2.10), (2.11) that we are looking for, in the case n < −1 and n 6= −3, is given by (2.12).

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Consider now the case n = −3. We can rewrite (2.20), (2.21) in the form (u0− u)2y′′− (u0− u) y′−₄1[1 + y − (_uu₀)σ(1 + y)−3] = 0, 0 < u < u0, y (u0) = lim u→u− 0 [(u0− u) y′(u)] = 0. (2.28)

In this case it is not possible to find a particular solution in the form (2.23), then following [16], we shall look for a one-parameter family of solutions of this problem in the form

y(u, b) = b (u0− u) [1 + b1(u0− u) + · · · ] + (u0− u) ln u0−u u0 [c0+ c1(u0− u) + · · · ] + (u0 − u)2ln2 u0−u u0 [d0+ d1(u0− u) + · · · ] + · · · ,

where b is the parameter and the coefficients bi, ci and di generally depending on b, may

be determined substituting y(u, b) in (2.28). So, in the case n = −3 the one-parameter family of solutions of the Cauchy problem (2.10), (2.11) we are looking for, is given by (2.13).

Consider now the case −1 < n < 0. If in the neighborhood of u = u0 we search for a

solution in the form

g (u) = C1(u0− u) + C2(u0− u)k[1 + o (1)], g′_{(u) = −C}1− kC2(u0− u)k−1[1 + o (1)], g′′_{(u) = k (k − 1) C}2(u0 − u)k−2[1 + o (1)], u → u−0. From (2.19), k (k − 1) C2(u0− u)k−2C1−n(u0− u)−n[1 + C2 C1 (u0− u)k−1]−n|_u→u− 0 = au σ 0,

which implies that k = 2 + n > 0, k − 1 = n + 1 > 0 and Cn 1 = C2 (2 + n) (n + 1) auσ 0 . With C2 = b < 0 we get C1 = b(2 + n) (n + 1) auσ 0 1 n . Next, we define the function y (u) by

g (u) = C1(u0− u) + b (u0− u)k[1 + y (u)].

Substituting in (2.10), (2.11), we obtain the following Cauchy problem in y (u0− u)2y′′− 2 (2 + n) (u0− u) y′+ (2 + n) (n + 1) [1 + y− u u0 σ 1 + b C1 (u0− u)n+1(1 + y) n = 0, 0 < u < u0, (2.29) y (u0) = lim u→u− 0 [(u0− u) y′(u)] = 0, (2.30)

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2. Singular Boundary Value Problems Over Bounded Domains where u = u0 is a regular singular point.

First, note that for any b 6= 0, this problem has a particular solution that can be represented by yp(u, b) = +∞ X l=0,k=0,l+k≥1 yl,k(b) (u0− u)l+k(1+n), 0 ≤ u0− u ≤ δ (b) , δ (b) ≥ 0,

where the coefficients yl,k may be determined substituting yp in (2.29), which gives

y1,0(b) = 0 and (2.15) in the case l = 0, k = 1.

Changing to the variable ˜_{y = y − y}p, (2.29), (2.30) becomes

(u0− u)2y˜′′− 2 (2 + n) (u0− u) ˜y′+ (2 + n) (n + 1) ˜y = f (u, ˜y), 0 < u < u0, (2.31)

˜

y (u0) = lim u→u−

0

[(u0− u) ˜y′(u)] = 0, (2.32)

whose characteristic exponents are λ1 = −1 − n < 0 and λ2 = −2 − n < 0 and

f (u, ˜y) = u u0 σ 1 + b C1 (u0− u)n+1(1 + ˜y + yp) n − 1 + b C1 (u0− u)n+1(1 + yp) n

satisfies f (u, 0) = 0. Moreover, since ∂f_∂u and ∂ ˜_{∂ ˜}f_y are continuous functions for every u0−δǫ<

u < u0 and |˜y| < ξǫ, by the mean value theorem, we have |f(u,˜y)−f(u,0)|_|˜_y−0| = |f(u,˜_|˜_y|y)| ≤ ǫ.

Performing the variable substitution t = u0 − u, we can apply corollary 1.1 to conclude

that problem (2.31), (2.32) has no other solution apart from ˜_{y(u) ≡ 0. Therefore, problem} (2.29), (2.30) has no other solution than yp(u, b).

It remains to analyze the case n = −1. If in the neighborhood of u = u0, we seek a

solution in the form

g (u) = C (u0− u) r − lnu0−u u0 [1 + o (1)], g′_{(u) = −C} "r − lnu0−u u0 − 2 r − lnu0−u u0 −1# [1 + o (1)], g′′_{(u) = −C} 2 (u0− u) r − lnu0−u u0 −1 1 −2 lnu0−u u0 −1 [1 + o (1)], as u → u−0, from (2.19) we conclude that C =

√ −2auσ

0. Next, we define y (u) by

g (u) = C (u0− u) s − ln u0− u u0 [1 + y (u)].