Nonlinear Second Order Elliptic Equations

✍ Scribed by Mingxin Wang, Peter Y. H. Pang

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 320
• Edition: 2024
• Category: Library

✦ Synopsis

This book focuses on the following three topics in the theory of boundary value problems of nonlinear second order elliptic partial differential equations and systems: (i) eigenvalue problem, (ii) upper and lower solutions method, (iii) topological degree method, and deals with the existence of solutions, more specifically non-constant positive solutions, as well as the uniqueness, stability and asymptotic behavior of such solutions.

While not all-encompassing, these topics represent major approaches to the theory of partial differential equations and systems, and should be of significant interest to graduate students and researchers. Two appendices have been included to provide a good gauge of the prerequisites for this book and make it reasonably self-contained.

A notable strength of the book is that it contains a large number of substantial examples. Exercises for the reader are also included. Therefore, this book is suitable as a textbook for graduate students who havealready had an introductory course on PDE and some familiarity with functional analysis and nonlinear functional analysis, and as a reference for researchers.

✦ Table of Contents

Preface
Contents
1 Preliminaries
1.1 Notations, Conventions and Basic Assumptions
1.2 Calculus in Banach Spaces
1.2.1 Fréchet Derivative
1.2.2 Gâteaux Derivative
1.3 Unconditional Local Extremum
1.4 Applications
Notes
Exercises
2 Eigenvalue Problems of Second Order Linear Elliptic Operators
2.1 Principal Eigenvalue of the Non-divergence Operator
2.1.1 The Existence and Uniqueness of Principal Eigenvalue
2.1.2 Equivalent Forms of the Maximum Principle
2.2 Eigenvalues of the Divergence Operator
2.2.1 Extremal Properties of Eigenvalues
2.2.2 The Totality of Eigenvalues and Eigenfunctions
2.2.3 Variation of Eigenvalues
2.2.4 Nodal Domains
2.2.5 Relations Between Principal Eigenvalue and Spectral Radius
2.3 The Eigenvalue Problem with Indefinite Weight Function
2.3.1 Dirichlet and Robin Boundary Conditions
2.3.2 Neumann Boundary Conditions
2.4 The Eigenvalue Problem of Non-complete Coupled System
2.5 Applications of the Completeness of Eigenfunctions
Notes
Exercises
3 Upper and Lower Solutions Method for Single Equations
3.1 Comparison Principle and Uniqueness of Positive Solutions
3.2 Upper and Lower Solutions Method
3.2.1 Existence of Solutions
3.2.2 Monotone Iterative Method
3.3 Existence and Uniqueness of Positive Solutions of Logistic Type Equation
3.4 Asymptotic Behavior of Positive Solution of Logistic Equation
3.5 Equations with Variable Coefficients and Degeneracy
3.5.1 Existence and Asymptotic Behavior of Positive Solution
3.5.2 Perturbation and Profile of Positive Solution
3.6 Equations with Nonlinear Boundary Conditions
3.7 Weak Upper and Lower Solutions Method
3.7.1 Semi-linear Equation
3.7.2 Quasi-linear Equation
3.8 Equations in Unbounded Domains
3.8.1 Upper and Lower Solutions Method
3.8.2 Approximation by Boundary Value Problems in Bounded Domains
Notes
Exercises
4 Upper and Lower Solutions Method for Systems
4.1 Weakly Coupled Systems
4.1.1 Existence of Solutions
4.1.2 Monotone Iterative Method
4.2 Existence of Positive Solutions of Weakly Coupled Systems
4.3 Strongly Coupled Elliptic Systems
4.3.1 Abstract Framework
4.3.2 Existence of Positive Solutions of Strongly Coupled System
4.4 Systems with Nonlinear Boundary Conditions
4.5 Systems in Unbounded Domains
Notes
Exercises
5 Theory of Topological Degree in Cones and Applications
5.1 Theory of Topological Degree in Cones
5.2 Relations Between Stability and Fixed Point Index
5.3 Calculation of Fixed Point Index and Existence of Positive Solutions
5.4 Positive Solutions of a Prey-Predator Model
5.4.1 A Priori Estimates and Fixed Point Indices
5.4.2 The Existence of Positive Solution
5.4.3 Stability, Multiplicity and Uniqueness of PositiveSolutions
5.5 A Diffusive Eco-Epidemiological Prey-Predator Model
5.5.1 A Priori Estimates
5.5.2 The Existence, Uniqueness and Non-degeneracy of Positive Solutions of (5.39)
5.5.3 The Existence, Uniqueness and Non-degeneracy of Positive Solutions of (5.40)
5.5.4 The Existence and Uniqueness of Positive Solutions of the Problem (5.37)
5.6 An Application of Theorem 5.3
Notes
Exercises
6 Systems with Homogeneous Neumann Boundary Conditions
6.1 Calculation of Index at the Positive Constant Solution
6.2 The Schnakenberg Model
6.2.1 Bounds for Positive Solutions
6.2.2 Non-existence of Non-constant Positive Solutions
6.2.3 Existence of Non-constant Positive Solutions
6.3 A Prey-Predator Model with Stage Structureand Cross-Diffusion
6.3.1 Bounds for Positive Solutions
6.3.2 Non-existence of Non-constant Positive Solutions
6.3.3 Existence of Non-constant Positive Solutions
Notes
Exercises
7 The p-Laplace Equations and Systems
7.1 Regularity, Maximum Principle and Harnack Inequality
7.2 The Eigenvalue Problem
7.2.1 The Principal Eigenvalue
7.2.2 The Eigenvalue Problem with Indefinite Weight Function
7.2.3 Properties of the Principal Eigenvalue
7.3 Comparison Principle
7.4 Weak Upper and Lower Solutions Method for the p-Laplace Equation
7.5 Applications
7.5.1 A Boundary Value Problem
7.5.2 A Nonlinear Eigenvalue Problem of p-Laplacian
7.6 Upper and Lower Solutions Method for the p-Laplace System
Notes
Exercises
A Basic Results of Sobolev Spaces and Nonlinear Functional Analysis
A.1 Some Properties of Wk, p(Ω)
A.2 Sobolev Inequalities
A.3 Imbedding Theorems
A.4 Basic Results of Nonlinear Functional Analysis
B Basic Theory of Elliptic Equations
B.1 Maximum Principle and Harnack Inequality
B.1.1 The Case of Classical Solution
B.1.2 The Case of Strong Solution
B.1.3 The Case of Weak Solution
B.2 Schauder Theory of Elliptic Equations
B.3 Lp Theory of Elliptic Equations
References
Index