 | Ghergu M. Nonlinear PDEs: mathematical models in biology, chemistry and population genetics / M.Ghergu, V.D.Radulescu. - Berlin; Heidelberg: Springer, 2012. - xviii, 391 p. - (Springer monographs in mathematics). - Bibliogr.: p.377-385. - Ind.: p.387-391. - ISBN 978-3-642-22663-2; ISSN 1439-7382
|
1 Overview of Mathematical Methods in Partial Differential
Equations .................................................... 1
1.1 Comparison Principles ................................... 1
1.2 Radial Symmetry of Solutions to Semilinear Elliptic
Equations ............................................... 6
1.3 Variational Methods ..................................... 9
1.3.1 Ekeland's Variational Principle .................. 9
1.3.2 Mountain Pass Theorem ........................... 11
1.3.3 Around the Palais-Smale Condition for Even
Functionals ..................................... 12
1.3.4 Bolle's Variational Method for Broken
Symmetries ...................................... 14
1.4 Degree Theory .......................................... 15
1.4.1 Brouwer Degree .................................. 15
1.4.2 Leray-Schauder Degree ........................... 16
1.4.3 Leray-Schauder Degree for Isolated Solutions .... 17
2 Liouville Type Theorems for Elliptic Operators
in Divergence Form .......................................... 19
2.1 Introduction ........................................... 19
2.2 Some Related ODE Problems .............................. 21
3 2.3 Main Results ........................................... 26
low-Up Boundary Solutions of the Logistic Equation .......... 29
3.1 Singular Solutions of the Logistic Equation ............ 30
3.1.1 A Karamata Regular Variation Theory Approach .... 43
3.2 Keller-Osserman Condition Revisited .................... 60
3.2.1 Setting of the Problem .......................... 61
3.2.2 Minimality Principle ............................ 65
3.2.3 Existence of Solutions on Some Ball ............. 70
3.2.4 Existence of Solutions on Small Balls ........... 72
3.2.5 Existence of Solutions on Smooth Domains ........ 73
3.2.6 Blow-Up Rate of Radially Symmetric Solutions .... 74
3.2.7 Blow-Up Rate of Solutions on Smooth Domains ..... 75
3.2.8 A Uniqueness Result ............................. 77
3.2.9 Discrete Equations .............................. 79
3.2.10 Numerical Computations .......................... 86
3.3 Entire Large Solutions ................................. 91
3.3.1 A Useful Result: Bounded Entire Solutions ....... 91
3.3.2 Existence of an Entire Large Solution ........... 93
3.3.3 Uniqueness of Solution .......................... 98
3.4 Elliptic Equations with Absorption .................... 100
3.5 Lack of the Keller-Osserman Condition ................. 106
4 Singular Lane-Emden-Fowler Equations and Systems ........... 117
4.1 Bifurcation Problems for Singular Elliptic Equations .. 117
4.2 Lane-Emden-Fowler Systems with Negative Exponents ..... 130
4.2.1 Preliminary Results ............................ 132
4.2.2 Nonexistence of a Solution ..................... 139
4.2.3 Existence of a Solution ........................ 142
4.2.4 Regularity of Solution ......................... 149
4.2.5 Uniqueness ..................................... 153
4.3 Sublinear Lane-Emden Systems with Singular Data ....... 155
4.3.1 Case p > 0 and q > 0 ........................... 155
4.3.2 Case p > 0 and q < 0 ........................... 158
4.3.3 Case p < 0 and q < 7 < 0 ....................... 162
4.3.4 Further Extensions: Superlinear Case ........... 163
5 Singular Elliptic Inequalities in Exterior Domains ......... 167
5.1 Introduction .......................................... 167
5.2 Some Elliptic Problems in Bounded Domains ............. 168
5.3 An Equivalent Integral Condition ...................... 174
5.4 The Nondegenerate Case ................................ 175
5.4.1 Nonexistence Results ........................... 175
5.4.2 Existence Results .............................. 180
5.5 The Degenerate Case ................................... 188
5.6 Application to Singular Elliptic Systems in Exterior
Domains ............................................... 203
6 Two Quasilinear Elliptic Problems .......................... 211
6.1 A Degenerate Elliptic Problem with Lack of
Compactness ........................................... 211
6.1.1 Introduction ................................... 211
6.1.2 Auxiliary Results .............................. 214
6.1.3 Proof of the Main Result ....................... 222
6.2 A Quasilinear Elliptic Problem for p-Laplace
Operator .............................................. 227
7 Some Classes of Polyharmonic Problems ...................... 245
7.1 An Eigenvalue Problem with Continuous Spectrum ........ 245
7.2 Infinitely Many Solutions for Perturbed
Nonlinearities ........................................ 251
7.3 A Biharmonic Problem with Singular Nonlinearity ....... 258
8 Large Time Behavior of Solutions for Degenerate Parabolic
Equations .................................................. 267
8.1 Introduction .......................................... 267
8.2 Superlinear Case ...................................... 268
8.3 Sublinear Case ........................................ 275
8.4 Linear Case ........................................... 283
9 Reaction-Diffusion Systems Arising in Chemistry ............ 287
9.1 Introduction .......................................... 287
9.2 Brusselator Model ..................................... 288
9.2.1 Existence of Global Solutions .................. 290
9.2.2 Stability of the Uniform Steady State .......... 293
9.2.3 Diffusion-Driven Instability ................... 295
9.2.4 A Priori Estimates ............................. 296
9.2.5 Nonexistence Results ........................... 299
9.2.6 Existence Results .............................. 302
9.3 Schnackenberg Model ................................... 306
9.3.1 The Evolution System and Global Solutions ...... 307
9.3.2 A Priori Estimates ............................. 310
9.3.3 Nonexistence of Nonconstant Steady States ...... 314
9.3.4 Existence Results .............................. 317
9.4 Lengyel-Epstein Model ................................. 322
9.4.1 Global Solutions in Time ....................... 323
9.4.2 Turing Instabilities ........................... 326
9.4.3 A Priori Estimates for Stationary Solutions .... 328
9.4.4 Nonexistence Results ........................... 330
9.4.5 Existence ...................................... 332
10 Pattern Formation and the Gierer-Meinhardt Model in
Molecular Biology .......................................... 337
10.1 Introduction .......................................... 337
10.2 Some Preliminaries .................................... 340
10.3 Case 0 ≤ p < 1 ........................................ 347
10.3.1 Existence ...................................... 347
10.3.2 Further Results on Regularity .................. 354
10.3.3 Uniqueness of a Solution ....................... 356
10.4 Case p < 0 ............................................ 362
10.4.1 A Nonexistence Result .......................... 362
10.4.2 Existence ...................................... 364
A Caffarelli-Kohn-Nirenberg Inequality ....................... 369
В Estimates for the Green Function Associated to the
Biharmonic Operator ........................................ 373
References ................................................. 377
Index ......................................................... 387
|
|
