Willem's book is devoted to minimax theorems and their applications to partial differential equations. Presenting basic minimax theorems in a simple and unified way, this book may serve as a textbook for advanced graduate students in partial differential equations. Booknews A textbook for an advanced graduate course in partial differential equations. Presents basic minimax theorems starting from a quantitative deformation lemma; and demonstrates their applications to partial differential equations, particularly in problems dealing with a lack of compactness. Includes some previously unpublished results such as a treatment of the generalized Kadomtsev-Petviashvili equation. Annotation c. Book News, Inc., Portland, OR (booknews.com) Cover......Page 1 Title page......Page 2 Introduction......Page 10 1.1 Differentiable functionals......Page 16 1.2 Quantitative deformation lemma......Page 20 1.3 Mountain pass theorem......Page 21 1.4 Semilinear Dirichlet problem......Page 23 1.5 Symmetry and compactness......Page 25 1.6 Symmetric solitary waves......Page 27 1.7 Subcritical Sobolev inequalities......Page 29 1.8 Non symmetric solitary waves......Page 32 1.9 Critical Sobolev inequality......Page 35 1.10 Critical nonlinearities......Page 41 2.1 Quantitative deformation lemma......Page 46 2.2 Ekeland variational principle......Page 48 2.3 General minimax principle......Page 50 2.4 Semilinear Dirichlet problem......Page 52 2.5 Location theorem......Page 58 2.6 Critical nonlinearities......Page 59 3.1 Equivariant deformation......Page 64 3.2 Fountain theorem......Page 65 3.3 Semilinear Dirichlet problem......Page 67 3.4 Multiple solitary waves......Page 69 3.5 A dual theorem......Page 74 3.6 Concave and convex nonlinearities......Page 76 3.7 Concave and critical nonlinearities......Page 77 4.2 Ground states......Page 80 4.3 Properties of critical values......Page 83 4.4 Nodal solutions......Page 85 5.1 Category......Page 90 5.2 Relative category......Page 91 5.3 Quantitative deformation lemma......Page 95 5.4 Minimax theorem......Page 98 5.5 Critical nonlinearities......Page 99 6.1 Degree theory......Page 104 6.2 Pseudogradient flow......Page 106 6.3 Generalized linking theorem......Page 109 6.4 Semilinear Schrödinger equation......Page 111 7.1 Definition of solitary waves......Page 118 7.2 Functional setting......Page 119 7.3 Existence of solitary waves......Page 121 7.4 Variational identity......Page 123 8.1 Invariance by translations......Page 126 8.2 Symmetric domains......Page 133 8.3 Invariance by dilations......Page 134 8.4 Symmetric domains......Page 141 Appendix A : Superposition operator......Page 142 Appendix B : Variational identities......Page 144 Appendix C : Symmetry of minimizers......Page 148 Appendix D : Topological degree......Page 154 Bibliography......Page 162 Index of Notations......Page 170 Index......Page 171
