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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
Front Cover for Morse Theoretic Aspects of $p$-Laplacian Type Operators
Available Formats:
Hardcover ISBN: 978-0-8218-4968-2
Product Code: SURV/161
141 pp 
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
141 pp 
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
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Front Cover for Morse Theoretic Aspects of $p$-Laplacian Type Operators
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  • Front Cover for Morse Theoretic Aspects of $p$-Laplacian Type Operators
  • Back Cover for Morse Theoretic Aspects of $p$-Laplacian Type Operators
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
141 pp 
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
141 pp 
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
    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.

    Readership

    Research mathematicians interested in nonlinear partial differential equations.

  • Table of Contents
     
     
    • 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
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Volume: 1612010
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.

Readership

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
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