Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.
Author(s): Antonio Ambrosetti, Andrea Malchiodi
Series: Cambridge studies in advanced mathematics 104
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 330
City: Cambridge, UK; New York
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Differential calculus......Page 15
1.2 Function spaces......Page 18
1.3 Nemitski operators......Page 19
1.4 Elliptic equations......Page 21
1.4.1 Eigenvalues of linear Dirichlet boundary value problems......Page 22
1.4.2 Regularity......Page 24
1.4.3 Positive solutions......Page 25
PART I Topological methods......Page 27
2.1 Bifurcation: definition and necessary conditions......Page 29
2.2 The Lyapunov--Schmidt reduction......Page 32
2.3 Bifurcation from the simple eigenvalue......Page 33
Applications to nonlinear eigenvalue problems......Page 37
3.1 Brouwer degree and its properties......Page 40
3.2 Application: the Brouwer fixed point theorem......Page 44
3.3 An analytic definition of the degree......Page 45
3.3.1 Degree for C2 maps......Page 46
3.3.2 Degree for continuous maps......Page 49
3.3.3 Properties of the degree......Page 50
3.4.1 Defining the Leray–Schauder degree......Page 52
3.5 The Schauder fixed point theorem......Page 57
3.6 Some applications of the Leray–Schauder degree to elliptic equations......Page 58
3.6.1 Sublinear problems......Page 59
3.6.2 Problems at resonance......Page 60
3.6.3 Exact multiplicity results......Page 63
3.7 The Krasnoselski bifurcation theorem......Page 66
3.8 Exercises......Page 68
4.1 Improving the homotopy invariance......Page 69
4.2 An application to a boundary value problem sub- and super-solutions......Page 71
4.3 The Rabinowitz global bifurcation theorem......Page 74
4.4 Bifurcation from infinity and positive solutions asymptotically linear elliptic problems......Page 79
4.5 Exercises......Page 87
PART II Variational methods, I......Page 89
5.1 Functionals and critical points......Page 91
5.2 Gradients......Page 92
5.3 Existence of extrema......Page 94
5.4 Some applications......Page 96
5.5 Linear eigenvalues......Page 100
5.6 Exercises......Page 102
6.1 Differentiable manifolds, an outline......Page 103
6.2 Constrained critical points......Page 107
6.3 Manifolds of codimension one......Page 109
6.4 Natural constraints......Page 111
6.5 Exercises......Page 112
7.1 Deformations of sublevels......Page 114
7.2 The steepest descent flow......Page 115
7.3 Deformations and compactness......Page 119
7.4 The Palais–Smale condition......Page 121
7.6 An application to a superlinear Dirichlet problem......Page 123
7.7 Exercises......Page 128
8.1 The mountain pass theorem......Page 130
8.1.1 Existence of pseudogradients......Page 136
8.2 Applications......Page 137
8.3 Linking theorems......Page 143
8.3.1 Linking and critical points......Page 146
8.4 The Pohozaev identity......Page 149
8.5 Exercises......Page 152
PART III Variational methods, II......Page 155
9.1 The Lusternik–Schnirelman category......Page 157
9.2 Lusternik–Schnirelman theorems......Page 161
9.3 Exercises......Page 169
10.1 The Krasnoselski genus......Page 171
10.2 Existence of critical points......Page 174
10.3 Multiple critical points of even unbounded functionals......Page 178
10.4 Applications to Dirichlet boundary value problems......Page 184
10.5 Exercises......Page 190
11.1 Radial solutions of semilinear elliptic equation on Rn......Page 191
11.2 Boundary value problems with critical exponent......Page 194
11.3.1 A general result......Page 202
11.3.2 A problem with multiple solutions......Page 208
11.4 Problems with concave-convex nonlinearities......Page 212
11.5 Exercises......Page 217
12.1.1 The axiomatic construction......Page 218
12.1.2 Singular homology groups......Page 223
12.1.3 Singular relative homology groups......Page 224
12.2 The Morse inequalities......Page 226
12.3 An application: bifurcation for variational operators......Page 238
12.4.1 An abstract result......Page 243
12.4.2 Some applications......Page 246
12.5 Exercises......Page 249
PART IV Appendices......Page 253
A1.1 The Gidas–Ni–Nirenberg symmetry result......Page 255
A1.2 A Liouville type theorem by Gidas and Spruck......Page 260
A1.3 An application......Page 262
A2.1 The abstract result......Page 266
A2.2 Semilinear elliptic equations on Rn......Page 268
A2.2.1 An existence result under assumption (a1)......Page 269
A2.2.2 An existence result under assumption (a2)......Page 272
A3.1 Bifurcation for problems on Rn in the presence of eigenvalues......Page 276
A3.2 Bifurcation from the essential spectrum......Page 283
A4.1 Formulation of the problem......Page 288
A4.2 Global existence results......Page 290
A4.3.1 A result by Fraenkel and Berger......Page 297
A4.3.2 Bifurcation from the Hill spherical vortex......Page 298
A5.1 An abstract result......Page 300
A5.2 Elliptic equations on Rn......Page 305
A5.3 Semiclassical states of nonlinear Schrödinger equations......Page 307
A5.4 Singularly perturbed Neumann problems......Page 310
A5.5 Perturbation of even functionals......Page 312
A6.1 The Yamabe problem......Page 316
A6.2 The scalar curvature problem......Page 319
References......Page 323
Index......Page 329
