Critical Point Theory and Its Applications

By: Wenming Zou, Martin Schechter

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Hardcover | 1 June 2006

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332 Pages

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23.5 x 15.24 x 2.54

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Since the birth of the calculus of variations, researchers have discovered that variational methods, when they apply, can obtain better results than most other methods. Moreover, they apply in a very large number of situations. It was realized many years ago that the solutions of a great number of problems are in effect critical points of functionals. Critical Point Theory and Its Applications presents some of the latest research in the area of critical point theory. Researchers have obtained many new results recently using this approach, and in most cases comparable results have not been obtained with other methods. This book describes the methods and presents the newest applications.

The topics covered in the book include extrema, even valued functionals, weak and double linking, sign changing solutions, Morse inequalities, and cohomology groups. The applications described include Hamiltonian systems, Schroedinger equations and systems, jumping nonlinearities, elliptic equations and systems, superlinear problems and beam equations. Many minimax theorems are established without the use of the (PS) compactness condition.

Industry Reviews

"Many, often difficult and advanced, examples included into the text form an excellent review of a frontline of actual research in the area. ... In conclusion, the reviewer may recommend the book of Zou and Schechter as an excellent reference for those seeking new as well as well-established techniques in the critical point theory approach to differential equations."

-Zentralblatt Math

"In many variational problems, the functional Phi is strongly indefinite and does not satisfy the Palais-Smale condition. In this book, the authors present some of the latest work which has been done to overcome these difficulties and prove the existence of critical points. They also show how the abstract results can be applied to many problems in ordinary and partial differential equations."

-Mathematical Reviews

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Preface	p. ix
Preliminaries	p. 1
Partition of Unity in Metric Spaces	p. 1
Sobolev Spaces	p. 2
Differentiable Functionals	p. 7
Topological Degrees	p. 13
An ODE in Banach Space	p. 16
The (PS) Conditions	p. 19
Weak Solutions	p. 20
Functionals Bounded Below	p. 25
Pseudo-Gradients	p. 25
Bounded Minimizing Sequences	p. 26
An Application	p. 31
Even Functionals	p. 37
Abstract Theorems	p. 37
High Energy Solutions	p. 44
Small Energy Solutions	p. 49
Linking and Homoclinic Type Solutions	p. 55
A Weak Linking Theorem	p. 55
Homoclinic Orbits of Hamiltonian Systems	p. 64
Asymptotically Linear Schrodinger Equations	p. 73
Schrodinger Equations with 0 [Element] Spectrum	p. 74
The Case of Critical Sobolev Exponents	p. 89
Schrodinger Systems	p. 101
The Superlinear Case	p. 102
The Asymptotically Linear Case	p. 112
Double Linking Theorems	p. 117
A Double Linking	p. 117
Twin Critical Points	p. 120
Eigenvalue Problems	p. 129
Jumping Nonlinearities	p. 131
Superlinear Problems	p. 141
Introduction	p. 141
Proofs	p. 145
The Eigenvalue Problem	p. 152
Systems with Hamiltonian Potentials	p. 159
A Linking Theorem	p. 159
Hamiltonian Elliptic Systems	p. 167
Linking and Elliptic Systems	p. 179
An Infinite-Dimensional Linking Theorem	p. 179
Elliptic Systems	p. 187
Sign-Changing Solutions	p. 195
Linking and Sign-Changing Solutions	p. 195
Free Jumping Nonlinearities	p. 202
Cohomology Groups	p. 215
The Kryszewski-Szulkin Theory	p. 215
Morse Inequalities	p. 227
The Shifting Theorem	p. 231
Critical Groups of Local Linking	p. 242
Computations of Cohomology Groups	p. 244
Hamiltonian Systems	p. 253
Asymptotically Linear Beam Equations	p. 269
Bibliography	p. 287
Index	p. 317
Table of Contents provided by Ingram. All Rights Reserved.

Critical Point Theory and Its Applications

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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