Singular Elliptic Problems: Bifurcation & Asymptotic Analysis (Math Applications Series)

✍ Scribed by Marius Ghergu, Vicentiu Radulescu

• Publisher: Oxford University Press, USA
• Year: 2008
• Tongue: English
• Leaves: 314
• Category: Library

✦ Synopsis

This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by singular elliptic equations. There are carefully analyzed logistic type equations with boundary blow-up solutions and generalized Lane-Emden-Fowler equations or Gierer-Meinhardt systems with singular nonlinearity in anisotropic media. These nonlinear problems appear as mathematical models in various branches of Physics, Mechanics, Genetics, Economics, Engineering, and they are also relevant in Quantum Physics and Differential Geometry.One of the main purposes of this volume is to deduce decay rates for general classes of solutions in terms of estimates of particular problems. Much of the material included in this volume is devoted to the asymptotic analysis of solutions and to the qualitative study of related bifurcation problems. Numerical approximations illustrate many abstract results of this volume. A systematic description of the most relevant singular phenomena described in these lecture notes includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity.The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear singular phenomena

✦ Table of Contents

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 5
Preface......Page 7
Acknowledgments......Page 11
Contents......Page 13
I PRELIMINARIES......Page 17
1.1 A fixed point result......Page 19
1.2 The method of sub- and supersolution......Page 21
1.3.1 Weak and strong maximum principle......Page 25
1.3.2 Maximum principle for weakly differentiable functions......Page 26
1.3.3 Stampacchia's maximum principle......Page 27
1.3.4 Vazquez's maximum principle......Page 28
1.3.5 Pucci and Serrin's maximum principle......Page 31
1.3.6 A comparison principle in the presence of singular nonlinearities......Page 33
1.4 Existence properties and related maximum principles......Page 36
1.4.1 Dead core solutions of sublinear logistic equations......Page 39
1.4.2 Singular solutions of the logistic equation......Page 40
1.5 Brezis-Oswald theorem -......Page 42
1.6 Comments and historical notes......Page 48
II BLOW-UP SOLUTIONS......Page 51
2.1 Introduction......Page 53
2.2.1 Existence results in bounded domains......Page 56
2.2.2 Existence results in the whole space......Page 58
2.3.1 The case of positive potentials......Page 61
2.3.2 The case of vanishing potentials......Page 62
2.4 An equivalent criterion to the Keller-Osserman condition......Page 65
2.5 Singular solutions of the logistic equation on domains with holes......Page 68
2.6 Uniqueness of blow-up solution......Page 74
2.7 A Karamata theory approach for uniqueness of blow-up solution......Page 80
2.8 Comments and historical notes......Page 87
3.1 Introduction......Page 91
3.2.1 Bounded or unbounded entire solutions......Page 92
3.2.2 Role of the Keller-Osserman condition......Page 97
3.3 Comments and historical notes......Page 105
III ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES......Page 107
4.1 Introduction......Page 109
4.2 An ODE with mixed nonlinearities......Page 114
4.3 A complete description for positive potentials......Page 115
4.4 An example......Page 120
4.5 Bifurcation for negative potentials......Page 123
4.6 Existence for large values of parameters in the sign-changing case......Page 128
4.7.1 Existence of entire solutions......Page 129
4.7.2 Uniqueness of radially symmetric solutions......Page 133
4.8 Comments and historical notes......Page 137
5 Bifurcation and asymptotic analysis: The monotone case......Page 141
5.1 Introduction......Page 142
5.2 A general bifurcation result......Page 143
5.3 Existence and bifurcation results......Page 145
5.4 Asymptotic behavior of the solution with respect to parameters......Page 150
5.5.1 First example......Page 153
5.5.2 Second example......Page 154
5.6 The case of singular nonlinearities......Page 156
5.7 Comments and historical notes......Page 158
6.1 Introduction......Page 159
6.2 Auxiliary results......Page 160
6.3 Existence and bifurcation results in the nonmonotone case......Page 163
6.5 Comments and historical notes......Page 170
7.1 Introduction......Page 173
7.2 The weak sub- and supersolution method......Page 174
7.3 H^1 local minimizers......Page 177
7.4 Existence of the first solution......Page 179
7.5 Existence of the second solution......Page 185
7.5.1 First case......Page 186
7.5.2 Second case......Page 190
7.6 C1 regularity of solution......Page 198
7.7 Asymptotic behavior of solutions......Page 200
7.8 Comments and historical notes......Page 205
8.1 Stability of the solution in a general singular setting......Page 207
8.2 A min-max characterization of the first eigenvalue for the linearized problem......Page 215
8.3 Differentiability of some singular nonlinear problems......Page 217
8.4 Examples......Page 220
8.5 Comments and historical notes......Page 221
9.1 Introduction......Page 223
9.2 A general nonexistence result......Page 224
9.3 A singular elliptic problem in one dimension......Page 226
9.4 Existence results in the sublinear case......Page 231
9.5 Existence results in the linear case......Page 242
9.6 Boundary estimates of the solution......Page 244
9.7 The case of a negative singular potential......Page 247
9.7.1 A nonexistence result......Page 248
9.7.2 Existence result......Page 249
9.8 Ground-state solutions of singular elliptic problems with gradient term......Page 254
9.9 Comments and historical notes......Page 257
10.1 Introduction......Page 259
10.2 A nonexistence result......Page 260
10.3 Existence results......Page 264
10.4 Uniqueness of the solution in one dimension......Page 271
10.5 Comments and historical notes......Page 277
A.1 Eigenvalues and eigenfunctions for the Laplace operator......Page 281
A.2 Krein-Rutman theorem......Page 282
Appendix B Implicit function theorem......Page 285
C.2 Ekeland's variational principle......Page 289
D.1 Ambrosetti-Rabinowitz theorem......Page 293
D.2 Application to the Emden-Fowler equation......Page 297
D.3 Mountains of zero altitude......Page 298
References......Page 299
Index......Page 311