An Introduction to Nonlinear Functional Analysis and Elliptic Problems 1st Edition by Antonio Ambrosetti, David Arcoya ISBN 0817681140 9780817681142

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ISBN 10: 0817681140
ISBN 13: 9780817681142
Author: Antonio Ambrosetti, David Arcoya

This self-contained textbook provides the basic, abstract tools used in nonlinear analysis and their applications to semilinear elliptic boundary value problems. By first outlining the advantages and disadvantages of each method, this comprehensive text displays how various approaches can easily be applied to a range of model cases. An Introduction to Nonlinear Functional Analysis and Elliptic Problems is divided into two parts: the first discusses key results such as the Banach contraction principle, a fixed point theorem for increasing operators, local and global inversion theory, Leray–Schauder degree, critical point theory, and bifurcation theory; the second part shows how these abstract results apply to Dirichlet elliptic boundary value problems. The exposition is driven by numerous prototype problems and exposes a variety of approaches to solving them. Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is a practical text for an introductory course or seminar on nonlinear functional analysis.

An Introduction to Nonlinear Functional Analysis and Elliptic Problems 1st Table of contents:

1 Preliminaries
1.1 Sobolev Spaces
1.1.1 Embedding Theorems
1.2 Linear Elliptic Equations
1.2.1 Fréchet Differentiability
1.2.2 Nemitski Operators
1.2.3 Dirichlet Principle
1.2.4 Regularity of the Solutions
1.2.5 The Inverse of the Laplace Operator
1.3 Linear Elliptic Eigenvalue Problems
1.3.1 Linear Compact Operators
1.3.2 Variational Characterization of The Eigenvalues
2 Some Fixed Point Theorems
2.1 The Banach Contraction Principle
2.2 Increasing Operators
3 Local and Global Inversion Theorems
3.1 The Local Inversion Theorem
3.2 The Implicit Function Theorem
3.3 The Lyapunov–Schmidt Reduction
3.4 The Global Inversion Theorem
3.5 A Global Inversion Theorem with Singularities
4 Leray–Schauder Topological Degree
4.1 The Brouwer Degree
4.2 The Leray–Schauder Topological Degree
4.2.1 Index of an Isolated Zero and Computation by Linearization
4.3 Continuation Theorem of Leray–Schauder
4.3.1 A Topological Lemma
4.3.2 A Theorem by Leray and Schauder
4.4 Other Continuation Theorems
5 An Outline of Critical Points
5.1 Definitions
5.2 Minima
5.3 The Mountain Pass Theorem
5.4 The Ekeland Variational Principle
5.5 Another Min–Max Theorem
5.6 Some Perturbation Results
6 Bifurcation Theory
6.1 Local Results
6.1.1 Bifurcation from a Simple Eigenvalue
6.1.2 Bifurcation from an Odd Eigenvalue
6.2 Bifurcation for Variational Operators
6.2.1 A Krasnoselskii Theorem for Variational Operators
6.2.2 Branching Points
6.3 Global Bifurcation
7 Elliptic Problems and Functional Analysis
7.1 Nonlinear Elliptic Problems
7.1.1 Classical Formulation
7.1.2 Weak Formulation
7.2 Sub- and Super-Solutions and Increasing Operators
8 Problems with A Priori Bounds
8.1 An Elementary Nonexistence Result
8.2 Existence of A Priori Bounds
8.3 Existence of Solutions
8.3.1 Using the Global Inversion Theorem
8.3.2 Using Degree Theory
8.3.3 Using Critical Point Theory
8.4 Positive Solutions
9 Asymptotically Linear Problems
9.1 Existence of Positive Solutions
9.2 Bifurcation from Infinity
9.3 On the Behavior of the Bifurcations from Infinity
9.4 The Local Anti-Maximum Principle
9.5 The Landesman–Lazer Condition
9.5.1 A Variational Proof of the Landesman–Lazer Result
10 Asymmetric Nonlinearities
10.1 The Approach by Ambrosetti and Prodi
10.2 The Approach by Amann–Hess
10.3 Variational Approach by Mountain Pass and Sub- and Super-Solutions
10.4 Approach by Degree Giving a Continuum of Solutions
11 Superlinear Problems
11.1 Using Min–Max Theorems
11.2 Superlinear Ambrosetti–Prodi Problem
12 Quasilinear Problems
12.1 First Results
12.2 Mountain Pass Theorem for Nondifferentiable Functionals and Applications
12.3 Application to Quasilinear Variational Problems
12.4 Some Nonvariational Quasilinear Problems
13 Stationary States of Evolution Equations
13.1 Soliton States to Stationary NLS Equations
13.2 Semiclassical States of NLS Equations with Potentials
13.3 Systems of NLS Equations
13.4 Nonautonomous Systems
13.5 Appendix
Appendix A
A.1 Weak Derivative
A.2 Sobolev Spaces
A.3 Boundary Values in Sobolev Spaces
A.4 Embedding Theorems

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