**Mathematical Surveys and Monographs**

Volume: 161;
2010;
141 pp;
Hardcover

MSC: Primary 58; 47; 35;

Print ISBN: 978-0-8218-4968-2

Product Code: SURV/161

List Price: $73.00

Individual Member Price: $58.40

**Electronic ISBN: 978-1-4704-1388-0
Product Code: SURV/161.E**

List Price: $73.00

Individual Member Price: $58.40

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#### Supplemental Materials

# Morse Theoretic Aspects of $p$-Laplacian Type Operators

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*Kanishka Perera; Ravi P. Agarwal; Donal O’Regan*

The purpose of this book is to present a Morse
theoretic study of a very general class of homogeneous operators that
includes the \(p\)-Laplacian as a special case. The
\(p\)-Laplacian operator is a quasilinear differential operator
that arises in many applications such as non-Newtonian fluid flows
and turbulent filtration in porous media. Infinite dimensional Morse
theory has been used extensively to study semilinear problems, but
only rarely to study the \(p\)-Laplacian.

The standard tools of Morse theory for computing critical groups,
such as the Morse lemma, the shifting theorem, and various linking and
local linking theorems based on eigenspaces, do not apply to
quasilinear problems where the Euler functional is not defined on a
Hilbert space or is not \(C^2\) or where there are no
eigenspaces to work with. Moreover, a complete description of the
spectrum of a quasilinear operator is generally not available, and the
standard sequence of eigenvalues based on the genus is not useful for
obtaining nontrivial critical groups or for constructing linking sets
and local linkings. However, one of the main points of this book is
that the lack of a complete list of eigenvalues is not an
insurmountable obstacle to applying critical point theory.

Working with a new sequence of eigenvalues that uses the
cohomological index, the authors systematically develop alternative
tools such as nonlinear linking and local splitting theories in order
to effectively apply Morse theory to quasilinear problems. They obtain
nontrivial critical groups in nonlinear eigenvalue problems and use
the stability and piercing properties of the cohomological index to
construct new linking sets and local splittings that are readily
applicable here.

#### Readership

Research mathematicians interested in nonlinear partial differential equations.

#### Table of Contents

# Table of Contents

## Morse Theoretic Aspects of $p$-Laplacian Type Operators

- Contents v6 free
- Preface vii8 free
- An Overview ix10 free
- Chapter 0. Morse Theory and Variational Problems 122 free
- Chapter 1. Abstract Formulation and Examples 1738
- Chapter 2. Background Material 2748
- Chapter 3. Critical Point Theory 4566
- Chapter 4. $p$-Linear Eigenvalue Problems 7192
- Chapter 5. Existence Theory 79100
- Chapter 6. Monotonicity and Uniqueness 87108
- Chapter 7. Nontrivial Solutions and Multiplicity 89110
- Chapter 8. Jumping Nonlinearities and the Dancer-Fucik Spectrum 97118
- Chapter 9. Indefinite Eigenvalue Problems 109130
- Chapter 10. Anisotropic Systems 117138
- Bibliography 135156