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Morse Theoretic Aspects of $p$-Laplacian Type Operators

Kanishka Perera Florida Institute of Technology, Melbourne, FL
Ravi P. Agarwal Florida Institute of Technology, Melbourne, FL
Donal O’Regan National University of Ireland, Galway, Ireland
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Hardcover ISBN: 978-0-8218-4968-2
Product Code: SURV/161
List Price: $78.00 MAA Member Price:$70.20
AMS Member Price: $62.40 Electronic ISBN: 978-1-4704-1388-0 Product Code: SURV/161.E List Price:$73.00
MAA Member Price: $65.70 AMS Member Price:$58.40
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List Price: $117.00 MAA Member Price:$105.30
AMS Member Price: $93.60 Click above image for expanded view Morse Theoretic Aspects of$p$-Laplacian Type Operators Kanishka Perera Florida Institute of Technology, Melbourne, FL Ravi P. Agarwal Florida Institute of Technology, Melbourne, FL Donal O’Regan National University of Ireland, Galway, Ireland Available Formats:  Hardcover ISBN: 978-0-8218-4968-2 Product Code: SURV/161  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$62.40
 Electronic ISBN: 978-1-4704-1388-0 Product Code: SURV/161.E
 List Price: $73.00 MAA Member Price:$65.70 AMS Member Price: $58.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$93.60
• Book Details

Mathematical Surveys and Monographs
Volume: 1612010; 141 pp
MSC: Primary 58; 47; 35;

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.

Research mathematicians interested in nonlinear partial differential equations.

• Chapters
• 1. Morse theory and variational problems
• 2. Abstract formulation and examples
• 3. Background material
• 4. Critical point theory
• 5. $p$-Linear eigenvalue problems
• 6. Existence theory
• 7. Monotonicity and uniqueness
• 8. Nontrivial solutions and multiplicity
• 9. Jumping nonlinearities and the Dancer-Fučík spectrum
• 10. Indefinite eigenvalue problems
• 11. Anisotropic systems

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1612010; 141 pp
MSC: Primary 58; 47; 35;

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.

Research mathematicians interested in nonlinear partial differential equations.

• Chapters
• 1. Morse theory and variational problems
• 2. Abstract formulation and examples
• 3. Background material
• 4. Critical point theory
• 5. $p$-Linear eigenvalue problems
• 6. Existence theory
• 7. Monotonicity and uniqueness
• 8. Nontrivial solutions and multiplicity
• 9. Jumping nonlinearities and the Dancer-Fučík spectrum
• 10. Indefinite eigenvalue problems
• 11. Anisotropic systems
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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