Boundary Value Problems for Quasilinear Second Order Differential Equations

Boundary Value Problems for Quasilinear Second Order Differential Equations

Abstract

This project is concerned with the review of some boundary value problems for non-linear ordinary dierential equations using topological and variational methods. A more classical boundary value problems for ordinary dierential equations (like the boundary value problems on a ball, initial value problems, problems on annular domains and positone problems) which represent the main interest of a wide number of researchers in the world is studied.

Chapter One

Introduction

The theory of di erential equations is a eld of mathematics that is motivated greatly by challenges arising from di erent applications, and leading to the birth of other elds of mathematics. It is not our intention to show a panoramic view of this enormous eld, we only intend to reveal its relation to the theory of dynamical systems.The mathematical results related to the investigation of di erential equation can be grouped as follows:

(a) Analytic and numerical methods for di erential equations.

(b) Prove the existence and uniqueness of solutions of di erential equations.

(c) Characterise the properties of solution without deriving explicit formulas for them. There is obviously a signi cant demand coming from application, for results in the rst direction. It is worthy to note that in the last century the emphasis was on the numerical approximation of solutions. The question of existence and uniqueness of initial value problem for ordinary di erential equations was answered completely in the rst half of the twentieth century ([43]), hence motivating the development of xed point theorems in normed spaces.

Today’s research is in the direction of existence and uniqueness of solutions of boundary value problems for non-linear ordinary di erential equations where the exact number of positive solutions are required is an actively studied eld . The studies in the third direction goes back to the end of the nineteenth century, when a considerable demand to investigate non-linear di erential equation appeared, and it turned out that these kind of equations cannot be solved analytically in most of the cases particularly as in the case of boundary value problems for quasilinear second order di erential equations.

In this work, we provide a survey of several results concerning solutions of quasilinear di erential equations where the independent variable vary over domains such as a ball, an annular domain determined by concentric spheres to determine if it has solutions which only depend upon the radial variable. This is well illustrated by the classical problems of nding the eigenvalues and the eigenfunctions of an operator subject to some boundary conditions on the domain of operation. Further problems are concerned with the existence of positive solutions of the equation

[ (u⁰)]⁰ + ^{N 1} (u⁰) + g( ; u) = 0; r 2 (a; b); u(a) = u(b) = 0 r

where > 0 and is a bounded domain in Rⁿ. In the case that = fx 2 Rⁿ : 0 < a < jxj < bg ; then the solutions of the above equation are solutions of the boundary value problem

u⁰⁰ + ^{N   1}u⁰ + f(u) = 0; r 2 (a; b); u(a) = u(b) = 0; (a; b)  :

r

This follows from the maximum principle for elliptic equations that solutions can only assume positive values in the interior of the domain. For N = 1; this problem is amenable to reduction of order methods, and are explicitly solved as demonstrated in section 3.3 illustrative example infra.

To establish that a given eigenvalue problem has positive solutions, we will start with the xed boundary conditions and try to nd the equation (by nding an appropriate coe – cient ) that has a nonzero solution satisfying the given boundary conditions. This kind of reversed boundary value problem is called an eigenvalue problem discussed in subsec-tion (4.3.2). The speci c value(s) of that would give a solution of the boundary value problem is called an eigenvalue of the boundary value problem. The nonzero solution that arises from each eigenvalue is called a corresponding eigenfunction of the boundary value problem. A typical example was given in ([45]) and discussed in chapter 3, section 3.3 of this work.