The maximum principle induces an order structure for partial differential equations, and has become an important tool in nonlinear analysis. This book is the first of two volumes to systematically introduce the applications of order structure in certain nonlinear partial differential equation problems. The maximum principle is revisited through the use of the Krein-Rutman theorem and the principal eigenvalues. Its various versions, such as the moving plane and sliding plane methods, are applied to a variety of important problems of current interest. The upper and lower solution method, especially its weak version, is presented in its most up-to-date form with enough generality to cater for wide applications. Recent progress on the boundary blow-up problems and their applications are discussed, as well as some new symmetry and Liouville type results over half and entire spaces. Some of the results included here are published for the first time.
Author(s): Yihong Du
Series: Partial Differential Equations and Application
Publisher: World Scientific Publishing Company
Year: 2006
Language: English
Pages: 202
Contents ......Page 10
Preface ......Page 6
1. Krein-Rutman Theorem and the Principal Eigenvalue ......Page 12
2.1 Equivalent forms of the maximum principle ......Page 20
2.2 Maximum principle in W2N(O) ......Page 22
3.1 Symmetry over bounded domains ......Page 28
3.2 Symmetry over the entire space ......Page 34
3.3 Positivity of nonnegative solutions ......Page 39
4.1 Classical upper and lower solutions ......Page 44
4.2 Weak upper and lower solutions ......Page 50
5.1 The classical case ......Page 72
5.2 The degenerate logistic equation ......Page 75
5.3 Perturbation and profile of solutions ......Page 86
6. Boundary Blow-Up Problems ......Page 94
6.1 The Keller-Osserman result and its generalizations ......Page 95
6.2 Blow-up rate and uniqueness ......Page 106
6.3 Logistic type equations with weights ......Page 113
7.1 Symmetry in a half space without strong maximum principle ......Page 128
7.2 Uniqueness results of logistic type equations over RN ......Page 139
7.3 Partial symmetry in the entire space ......Page 150
7.4 Some Liouville type results ......Page 156
A.l Schauder theory for elliptic equations ......Page 174
A.2 Sobolev spaces ......Page 177
A.3 Weak solutions of elliptic equations ......Page 181
A.4 LP theory of elliptic equations ......Page 185
A.5.1 The classical maximum principles ......Page 188
A.5.2 Maximum principles and Harnack inequality for weak solutions ......Page 189
A.5.3 Maximum principles and Harnack inequality for strong solutions ......Page 190
Bibliography ......Page 192
Index ......Page 200
