Minimax Systems and Critical Point Theory (eBook)

Contents 6
Preface 10
Critical Points of Functionals 14
1.1 Introduction 14
1.2 Extrema 15
1.3 PalaisÒSmale sequences 15
1.4 Cerami sequences 16
1.5 Linking sets 16
1.6 Previous definitions of linking 17
1.7 Notes and remarks 18
Minimax Systems 19
2.1 Introduction 19
2.2 Definitions and theorems 20
2.3 Linking subsets 21
2.4 A variation 23
2.5 Weaker conditions 24
2.6 Some consequences 25
2.7 Notes and remarks 27
Examples of Minimax Systems 28
3.1 Introduction 28
3.2 A method using homeomorphisms 28
3.3 A method using metric spaces 30
3.4 A method using homotopy-stable families 30
3.5 Examples of linking sets 32
3.6 Various geometries 35
3.7 A sandwich theorem 37
3.8 Notes and remarks 40
Ordinary Differential Equations 41
4.1 Extensions of PicardÌs theorem 41
4.2 Estimating solutions 43
4.3 Extending solutions 44
4.4 The proofs 45
4.5 An important estimate 46
The Method Using Flows 48
5.1 Introduction 48
5.2 Theorem 2.4 48
5.3 Theorem 2.12 50
5.4 Theorem 2.14 53
5.5 Theorem 2.21 54
Finding Linking Sets 60
6.1 Introduction 60
6.2 The strong case 61
6.3 The remaining proofs 63
6.4 Notes and remarks 65
Sandwich Pairs 66
7.1 Introduction 66
7.2 Criteria 67
7.3 Notes and remarks 70
Semilinear Problems 71
8.1 Introduction 71
8.2 Bounded domains 71
8.3 Some useful quantities 77
8.4 Unbounded domains 79
8.5 Further applications 83
8.6 Special cases 88
8.7 The proofs 89
8.8 Notes and remarks 91
Superlinear Problems 92
9.1 Introduction 92
9.2 The main theorems 93
9.3 Preliminaries 95
9.4 Proofs 96
9.5 The parameter problem 99
9.6 The monotonicity trick 104
9.7 Notes and remarks 111
Weak Linking 113
10.1 Introduction 113
10.2 Another norm 114
10.3 Some examples 118
10.4 Some applications 120
10.5 Notes and remarks 132
Fucik Spectrum: Resonance 134
11.1 Introduction 134
11.2 The curves 136
11.3 Existence 140
11.4 Notes and remarks 144
Rotationally Invariant Solutions 145
12.1 Introduction 145
12.2 The spectrum of the linear operator 146
12.3 The nonlinear case 148
12.4 Notes and remarks 151
Semilinear Wave Equations 152
13.1 Introduction 152
13.2 Convexity and lower semi-continuity 152
13.3 Existence of saddle points 155
13.4 Criteria for convexity 158
13.5 Partial derivatives 159
13.6 The theorems 161
13.7 The proofs 161
13.8 Notes and remarks 164
Type (II) Regions 165
14.1 Introduction 165
14.2 The asymptotic equation 168
14.3 Local estimates 170
14.4 The solutions 174
14.5 Notes and remarks 177
Weak Sandwich Pairs 178
15.1 Introduction 178
15.2 Weak sandwich pairs 179
15.3 Applications 185
15.4 Notes and remarks 191
Multiple Solutions 192
16.1 Introduction 192
16.2 Two examples 192
16.3 Statement of the theorems 193
16.4 Some lemmas 195
16.5 Local linking 207
16.6 The proofs 209
16.7 Notes and remarks 210
Second-Order Periodic Systems 213
17.1 Introduction 213
17.2 Proofs of the theorems 215
17.3 Nonconstant solutions 221
17.4 Notes and remarks 226
Bibliography 228
Index 239