Singular Elliptic Problems: Bifurcation & Asymptotic Analysis (Math Applications Series)

314 Pages·2008·3.907 MB·English

by  Marius Ghergu, Vicentiu Radulescu

OXIORD [tCTURS IN tlAi111WAYIC.1
AND ITS ArrIICATIONS 17
Singular
Elliptic
Problems
Bifurcation and Asymptotic Analysis
Marius Ghergu
Viceniiu RAdulescu
OXFORD LECTURE SERIES
IN MATHEMATICS AND ITS APPLICATIONS
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37. Ghergu and Radulescu, Singular Elliptic Problems: Bifurcation and
Asymptotic Analysis
Singular Elliptic Problems:
Bifurcation and Asymptotic Analysis
Marius Ghergu
Vicentiu D. RAdulescu
CLARENDON PRESS OXFORD
2008
OXFORD
UNIVERSITY PRESS
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Library of Congress Cataloging-in-Publication Data
Ghergu. Marius
Singular elliptic problems. bifurcation and asymptotic analysis I
Marius Ghergu, Viccntiu D. RSdulescu.
p. cm. - (Oxford lecture series in mathematics and its applications; 37)
Includes bibliographical references and index.
ISBN 978-0-19-533472-2
1. Differential equations, Elliptic-Asymptotic theory.
2. Differential equations, Nonlinear. 3. Bifurcation theory.
1. Radulescu, V. II. Title.
QA377.G47 2008
515`.3533-dc22 2007060129
135798642
Printed in the United States of America
on acid-free paper
To our families,
for their patience and continuous support over the years

PREFACE
The most incomprehensible thing
about the world is that it is
comprehensible.
Albert Einstein (1879-1955)
The development of nonlinear analysis during the last few decades has been
profoundly influenced by attempts to understand various phenomena from math-
ematical physics. One of the beauties of the subject is the immense breadth of
mathematics that has been applied in this pursuit.
There is an enormous body of literature in nonlinear elliptic partial differen-
tial equations that stretches back half a century. However, we shall make almost
no reference to this literature, and shall rely almost entirely upon personal re-
sults. These lecture notes are primarily intended to fill, in a substantial way,
the absence of a book dealing with the qualitative analysis of some basic singu-
lar stationary processes arising in nonlinear sciences. This volume aims to offer
an introduction to this subject, and also to present some research problems.
The models that we analyze represent a compromise between the description of
physical phenomena and analytical requirements; accordingly, our presentation is
characterized by a strict interplay between mathematics and nonlinear sciences.
The book is an outgrowth of our original research on the subject during the
last few years, and much of the development is motivated by problems arising
in applications. However, most of the proofs have been completely reworked and
we are especially careful to explain where each chapter is going, why it matters,
and what background material is required. Although the theory that we describe
could have been carried out on differentiable manifolds even from the beginning,
we have chosen to develop it on domains on the Euclidean space. However,
the techniques we develop can be extended to Laplace-Beltrami operators on
Riemannian manifolds.
The major thrust of this book is the qualitative analysis of some classes of
nonlinear stationary problems involving different types of singularities. Be aware,
this is definitely a research book. We are mainly concerned with the following
types of problems. We first study singular solutions of the logistic equation, with
a basic model that is described by the semilinear elliptic equation Du = uP,
where p > 1. The research program around this equation flourished after the
pioneering papers by Bieberbach and Rademacher, continued with the deep con-
tributions of Loewner and Nirenberg in Riemannian geometry, and creating re-
cently (because of the works by Dynkin and Le Gall) a nonlinear analogue of
the classical relation between Brownian motion and potential theory. Equations
of this type arise in astrophysics, genetics, meteorology, theory of atomic spec-
viii Preface
tra, and the Yamabe problem in geometry. A first consequence of such types
of nonlinearities is the possibility of the existence of a "large solution"-that
is, a solution blowing up at the boundary. When the large solution is unique,
it is a maximal solution and dominates any solution. In connection with the
previously mentioned applications, the existence of the large solution in a ball
was used by Iscoe to establish the compact support property of super-Brownian
motion, demonstrating the importance of the relationship between properties
of superdiffusion and the equation. Next, we are concerned with Lane-Emden-
Fowler equations and Gierer-Meinhardt systems with singular nonlinearity. The
model problem in such cases is described by equations like --Au = u-°, where
a is a positive real number. To our best knowledge, the first study in this di-
rection is from Fulks and Maybee, who proved existence and uniqueness results
by using a fixed point argument; moreover, they showed that solutions of the
associated parabolic problem tend to the unique solution of the corresponding
elliptic equation. Different approaches are the result of Coclite and Palmieri,
respectively Crandall, Rabinowitz, and Tartar, who approximated the singular
equation with regular problems, where the standard monotonicity techniques do
work. Singular problems of this type arise in the context of chemical heteroge-
neous catalysts and chemical catalyst kinetics, in the theory of heat conduction
in electrically conducting materials, singular minimal surfaces, as well as in the
study of non-Newtonian fluids, boundary layer phenomena for viscous fluids,
glacial advance, transport of coal slurries down conveyor belts, and in several
other geophysical and industrial contents. In both cases, because of the meaning
of the unknowns (concentrations, populations, etc.), the positive solutions are
relevant in most situations.
We intend to give a systematic treatment of the basic mathematical the-
ory and constructive methods for these classes of nonlinear elliptic equations, as
well as their applications to various processes arising in mathematical physics.
Our approach leads not only to the basic results of existence, uniqueness, and
multiplicity of solutions, but also to several qualitative properties, including bi-
furcation, asymptotic analysis, blow-up and so forth. Moreover, because the book
is concerned primarily with classical solutions, the monotone iteration processes
we apply for various classes of nonlinear singular problems are adaptable to nu-
merical solutions of the corresponding discrete processes. To place the text in
better perspective, each chapter is concluded with a section on historical notes
that includes references to all important and relatively new results. In addition
to cited works, the list of references contains many other works related to the
material developed in this volume.
The organization of the book is briefly summarized as follows. The first chap-
ter deals with preliminary material, such as the method of sub- and supersolu-
tion, several variants of the maximum principle (Stampacchia, Vazquez, Pucci,
and Serrin), and various existence and uniqueness results for nonlinear elliptic
boundary value problems.
Preface ix
Part II is composed of two chapters, which are concerned with singular so-
lutions of logistic-type equations or systems. There are studied both equations
with blow-up boundary solutions and entire solutions blowing up at infinity for
elliptic systems. In all these cases, the major role played by the Keller-Osserman
condition is discussed.
In the third part of this book we are concerned with elliptic problems involv-
ing singular nonlinearities, either in isotropic or in anisotropic media. Chapter 4
deals with sublinear elliptic problems that are affected by singular perturba-
tions. We distinguish between equations on bounded domains or on the whole
space and we are also concerned with a related bifurcation problem. Chapters 5
and 6 are devoted to the study of a bifurcation problem in the case of linear
growth for the nonlinearity. Two different situations are distinguished and a
complete discussion is developed in both circumstances. The superlinear case is
studied in Chapter 7, by means of variational arguments, whereas Chapter 8
is concerned with stability properties of solutions. Chapter 9 is devoted to the
study of the "competition" between various terms in a singular Lane-Emden-
Fowler equation with convection and variable (possible, singular) potential. In
the last chapter of these lecture notes, the qualitative analysis of solutions is ex-
tended to the case of singular Gierer-Meinhardt systems. We refer to the works
of J.M. Ball [11,12], V. Barbu [19], L. Beznea and N. Boboc [22], H. Brezis [30],
and P.G. Ciarlet [46,47] for related results and various applications to concrete
phenomena.
Four appendices illustrate some basic mathematical tools applied in this
book: elements of spectral theory for differential operators, the implicit func-
tion theorem, Ekeland's variational principle, and the mountain pass theorem.
These auxiliary chapters deal with some analytical methods used in this volume,
but also include some complements.
Each problem we develop in this book has its own difficulties. That is why
we intend to develop some standard and appropriate methods that are useful
and that can be extended to other problems. However, we do our best to re-
strict the prerequisites to the essential knowledge. We define as few concepts as
possible and give only basic theorems that are useful for our topic. The only
prerequisite for this volume is a standard graduate course in partial differential
equations, drawing especially from linear elliptic equations to elementary varia-
tional methods, with a special emphasis on the maximum principle (weak and
strong variants). This volume may be used for self-study by advanced graduate
students and engineers, and as a valuable reference for researchers in pure and
applied mathematics and physics.
Our vision throughout this volume is closely inspired by the following words
of Henri Poincare on the role of partial differential equations in the development
of other fields of mathematics and in applications: Nevertheless, each time I
can, I aim the absolute rigor for two reasons. In the first place, it is always
hard for a geometer to consider a problem without resolving it completely. In
the second place, these equations that I will study are susceptible, not only to
x Preface
physical applications, but also to analytical applications. It is using the existence
theory of the Dirichlet problem that Riemann founded his magnificent theory of
Abelian functions. Since then, other geometers have made important applications
of the same principle to the most fundamental parts of pure analysis. Is it still
permitted to content oneself with a demi-rigor? And who will say that the other
problems of mathematical physics will not, one day, be called to play in analysis
a considerable role, as has been the case of the most elementary of them? (Henri
Poincare 1164])
May 2007