Critical Point Theory - Sandwich and Linking Systems

Description

This monograph collects cutting-edge results and techniques for solving nonlinear partial differential equations using critical points. Including many of the author’s own contributions, a range of proofs are conveniently collected here, Because the material is approached with rigor, this book will serve as an invaluable resource for exploring recent developments in this active area of research, as well as the numerous ways in which critical point theory can be applied. Different methods for finding critical points are presented in the first six chapters. The specific situations in which these methods are applicable is explained in detail. Focus then shifts toward the book’s main subject: applications to problems in mathematics and physics. These include topics such as Schrödinger equations, Hamiltonian systems, elliptic systems, nonlinear wave equations, nonlinear optics, semilinear PDEs, boundary value problems, and equations with multiple solutions. Readers will find this collection of applications convenient and thorough, with detailed proofs appearing throughout. Critical Point Theory will be ideal for graduate students and researchers interested in solving differential equations, and for those studying variational methods. An understanding of fundamental mathematical analysis is assumed. In particular, the basic properties of Hilbert and Banach spaces are used. The Purpose of the Book......Page 6 A One-Dimensional Problem......Page 7 General Critical Points......Page 12 Linking......Page 13 Sandwich Pairs......Page 14 The Monotonicity Trick......Page 15 Weak Linking......Page 16 Differential Equations......Page 18 Applications......Page 19 Conclusion......Page 31 Contents......Page 32 1.1 Linking Pairs......Page 36 1.2 Flows......Page 39 1.3 A General Linking Theorem......Page 40 1.4 Linking Sets......Page 49 1.5 Various Geometries......Page 52 1.6 Notes and Remarks......Page 54 2.1 Sandwich Pairs......Page 56 2.2 Some Criteria......Page 57 2.3 Sandwich Systems......Page 60 2.4 Q Sandwich Pairs......Page 61 2.5 Notes and Remarks......Page 64 3.1 Using Linking......Page 65 3.2 Examples......Page 73 3.3 Notes and Remarks......Page 80 4.1 The Basic Problem......Page 81 4.2 Sandwich Systems......Page 82 4.3 Linking Systems......Page 83 4.4 The Parameter Problem......Page 85 4.5 Finding the Sequences......Page 87 4.6 The Monotonicity Trick......Page 93 4.7 Notes and Remarks......Page 94 5.1 Infinite Dimensional Splitting......Page 95 5.2 Weak Sandwich Systems......Page 98 5.3 Weak Linking Systems......Page 106 5.4 The Parameter Problem......Page 108 5.5 Finding the Sequences......Page 110 5.6 Notes and Remarks......Page 115 6.1 Extensions of Picard's Theorem......Page 116 6.2 Extending Solutions......Page 118 6.3 Maximum Intervals......Page 119 6.4 Upper and Lower Estimates......Page 122 6.5 Comparison of Solutions......Page 123 6.6 Summary......Page 125 6.7 Notes and Remarks......Page 126 7.2 Superlinear Problems......Page 127 7.3 The Schrödinger Equation......Page 129 7.4 Some Lemmas......Page 131 7.5 Proofs of the Theorems......Page 141 7.6 Notes and Remarks......Page 143 8.1 Introduction......Page 144 8.2 Zero a Boundary Point of ρ(A)......Page 145 8.3 The Schrödinger Equation......Page 147 8.4 Some Lemmas......Page 148 8.5 Proof of Theorem 8.1......Page 158 8.6 Notes and Remarks......Page 160 9.2 The Hypotheses......Page 162 9.3 The Space N1......Page 165 9.4 No Negative Eigenvalues......Page 166 9.5 Only One Negative Eigenvalue......Page 167 9.6 Two or More Negative Eigenvalues......Page 168 9.7 Least Energy Solutions......Page 169 9.9 Proof of Lemma 9.1......Page 170 9.10 Proofs of the Theorems......Page 171 9.11 Radial Solutions......Page 180 9.12 No Negative Eigenvalues......Page 182 9.13 Only One Negative Eigenvalue......Page 183 9.14 Two or More Negative Eigenvalues......Page 184 9.15 Least Energy Solutions......Page 185 9.16 The Operator A......Page 186 9.17 Some Lemmas......Page 188 9.19 Notes and Remarks......Page 195 10.1 Introduction......Page 197 10.2 The Case λ0 ≤0......Page 199 10.3 The Case λ0 >0......Page 201 10.4 The Operator D......Page 203 10.5 Proofs of the Theorems......Page 209 10.6 Some Lemmas......Page 214 10.7 The Remaining Proofs......Page 215 10.8 Notes and Remarks......Page 218 11.1 Use of Core Functions......Page 221 11.2 Boundary Value Problems......Page 223 11.3 Examples......Page 225 11.4 Lemmas......Page 226 11.5 Nontriviality......Page 230 11.6 Proofs of the Theorems......Page 231 11.7 Superlinear Problems......Page 237 11.8 Notes and Remarks......Page 240 12.1 Introduction......Page 242 12.2 Custom Flows......Page 244 12.3 Custom Sandwich Systems......Page 245 12.4 The Custom Parameter Problem......Page 246 12.5 Some Applications......Page 247 12.6 Finding the Sequences......Page 251 12.7 Notes and Remarks......Page 253 13.1 Introduction......Page 254 13.2 The Cooperative System......Page 255 13.3 Some Lemmas......Page 256 13.4 Proof of Theorem 13.1......Page 258 13.5 The Noncooperating System......Page 260 13.6 Contrast......Page 269 13.7 Notes and Remarks......Page 271 14.2 Flows......Page 272 14.3 Linking......Page 274 14.4 Cerami Sequences......Page 275 14.5 Some Applications......Page 278 14.6 Notes and Remarks......Page 282 15.1 Introduction......Page 283 15.2 The Spectrum of the Linear Operator......Page 284 15.3 Proof of Theorem 15.1......Page 286 15.4 Notes and Remarks......Page 288 16.1 Introduction......Page 289 16.2 Some Lemmas......Page 290 16.3 Proof of Theorem 16.1......Page 297 16.4 Multiple Lattices......Page 301 16.5 Notes and Remarks......Page 303 17.1 Introduction......Page 305 17.2 The Spectrum of the Linear Operator......Page 307 17.3 The Nonlinear Case......Page 309 17.4 Notes and Remarks......Page 318 18.2 Variational Setting and Preliminary Results......Page 320 18.3 Statement of the Theorems......Page 324 18.4 Proofs of the Theorems......Page 325 18.5 The Remaining Proofs......Page 327 18.6 Some Examples......Page 330 18.7 Notes and Remarks......Page 333 Bibliography......Page 334 Index......Page 346