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Sign-Changing Critical Point Theory

Sign-Changing Critical Point Theory
Author
Zou, Wenming 
Publisher
Springer-Verlag 
Publication Date
Sep, 2008 
ISBN
038776657X or 9780387766577
HARDCOVER 
278 Pages
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¥ 21,600 (tax included)

Description

Many nonlinear problems in physics, engineering, biology and social sciences can be reduced to finding critical points of functionals. While minimax and Morse theories provide answers to many situations and problems on the existence of multiple critical points of a functional, they often cannot provide much-needed additional properties of these critical points. Sign-changing critical point theory has emerged as a new area of rich research on critical points of a differentiable functional with important applications to nonlinear elliptic PDEs. Key features of this book: (1) Self-contained in-depth treatment of sign-changing critical point theory; (2) Further explorations in Minimax and Morse theories; (3) Topics devoted to linking and nodal solutions, the sign-changing saddle point theory, the generalized Brezis Nirenberg critical point theorem, the parameter dependence of sign-changing critical points; (3) Applications of sign-changing critical point theory studied within the classical symmetric mountain pass theorem; (4)Applies sign-changing concepts to Schrodinger equations and boundary value problems. This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis.

Contents

1 Preliminaries
1.1 Partition of Unity
1.2 Ekeland's Variational Principle
1.3 Sobolev Spaces and Embedding Theorems
1.4 Differentiable Functionals
1.5 The Topological Degree
1.6 The Global Flow
1.7 The Local Flow
1.8 The (PS) Condition
1.9 Lax- Milgram Theorem and Weak Solutions

2 Schechter- Tintarev Linking
2.1 Schechter- Tintarev Linking
2.2 Sign-Changing Critical Points via Linking
2.3 Jumping Dirichlet Equations
2.4 Oscillating Dirichlet Equations
2.5 Double Resonant Cases

3 Sign-Changing Saddle Point
3.1 Rabinowitz's Saddle Points
3.2 Sign-Changing Saddle Points
3.3 Schrodinger Equations with Potential Well
3.4 Flow-Invariant Sets
3.5 Sign-Changing Homoclinic-Type Solutions

4 On a Brezis- Nirenberg Theorem.
4.1 Introduction
4.2 Generalized Brezis- Nirenberg Theorems
4.3 Schrodinger Equations

5 Even Functionals
5.1 Introduction
5.2 An Abstract Theorem
5.3 A Classical Superlinear Problem
5.4 Composition Convergence Lemmas
5.5 Improved Hardy- Poincar´e Inequalities
5.6 Equations with Critical Hardy Constant
5.7 Critical Sobolev- Hardy Exponent Cases

6 Parameter Dependence
6.1 Bounded Sign-Changing (PS)-Sequences
6.2 Bounded (PS) Sequences via Symmetry
6.3 Positive and Negative Solutions
6.4 Subcritical Schrodinger Equation
6.5 Critical Cases

7 On a Bartsch- Chang- Wang-Weth Theory
7.1 Some Basic Results on Morse Theory
7.2 Critical Groups of Sign-Changing Critical Points
7.3 Sign-Changing Solutions of Mountain Pass Type
7.4 Nodal Domains
7.5 Sign-Changing Solutions with Least Energy

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