Singular Elliptic Problems Bifurcation &

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

Marius Ghergu is Researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy. Vicentiu Radulescu is Senior Researcher at the Institute of Mathematics "Simion Stoilow" of the Romanian Academy and Full Professor at the University of Craiova.

Preliminaries
Basic methods	p. 3
A fixed point result	p. 3
The method of sub- and supersolution	p. 5
Comparison principles	p. 9
Weak and strong maximum principle	p. 9
Maximum principle for weakly differentiable functions	p. 10
Stampacchia's maximum principle	p. 11
Vazquez's maximum principle	p. 12
Pucci and Serrin's maximum principle	p. 15
A comparison principle in the presence of singular nonlinearities	p. 17
Existence properties and related maximum principles	p. 20
Dead core solutions of sublinear logistic equations	p. 23
Singular solutions of the logistic equation	p. 24
Brezis-Oswald theorem	p. 26
Comments and historical notes	p. 32
Blow-up Solutions
Blow-up solutions for semilinear elliptic equations	p. 37
Introduction	p. 37
Blow-up solution for elliptic equations with vanishing potential	p. 40
Existence results in bounded domains	p. 40
Existence results in the whole space	p. 42
Blow-up solutions for logistic equations	p. 45
The case of positive potentials	p. 45
The case of vanishing potentials	p. 46
An equivalent criterion to the Keller-Osserman condition	p. 49
Singular solutions of the logistic equation on domains with holes	p. 52
Uniqueness of blow-up solution	p. 58
A Karamata theory approach for uniqueness of blow-up solution	p. 64
Comments and historical notes	p. 71
Entire solutions blowing up at infinity for elliptic systems	p. 75
Introduction	p. 75
Characterization of the central value set	p. 76
Bounded or unbounded entire solutions	p. 76
Role of the Keller-Osserman condition	p. 81
Comments and historical notes	p. 89
Elliptic Problems with Singular Nonlinearities
Sublinear perturbations of singular elliptic problems	p. 93
Introduction	p. 93
An ODE with mixed nonlinearities	p. 98
A complete description for positive potentials	p. 99
An example	p. 104
Bifurcation for negative potentials	p. 107
Existence for large values of parameters in the sign-changing case	p. 112
Singular elliptic problems in the whole space	p. 113
Existence of entire solutions	p. 113
Uniqueness of radially symmetric solutions	p. 117
Comments and historical notes	p. 121
Bifurcation and asymptotic analysis: The monotone case	p. 125
Introduction	p. 126
A general bifurcation result	p. 127
Existence and bifurcation results	p. 129
Asymptotic behavior of the solution with respect to parameters	p. 134
Examples	p. 137
First example	p. 137
Second example	p. 138
The case of singular nonlinearities	p. 140
Comments and historical notes	p. 142
Bifurcation and asymptotic analysis: The nonmonotone case	p. 143
Introduction	p. 143
Auxiliary results	p. 144
Existence and bifurcation results in the nonmonotone case	p. 147
An example	p. 154
Comments and historical notes	p. 154
Superlinear perturbations of singular elliptic problems	p. 157
Introduction	p. 157
The weak sub- and supersolution method	p. 158
H[superscript 1] local minimizers	p. 161
Existence of the first solution	p. 163
Existence of the second solution	p. 169
First case	p. 170
Second case	p. 174
C[superscript 1] regularity of solution	p. 182
Asymptotic behavior of solutions	p. 184
Comments and historical notes	p. 189
Stability of the solution of a singular problem	p. 191
Stability of the solution in a general singular setting	p. 191
A min-max characterization of the first eigenvalue for the linearized problem	p. 199
Differentiability of some singular nonlinear problems	p. 201
Examples	p. 204
Comments and historical notes	p. 205
The influence of a nonlinear convection term in singular elliptic problems	p. 207
Introduction	p. 207
A general nonexistence result	p. 208
A singular elliptic problem in one dimension	p. 210
Existence results in the sublinear case	p. 215
Existence results in the linear case	p. 226
Boundary estimates of the solution	p. 228
The case of a negative singular potential	p. 231
A nonexistence result	p. 232
Existence result	p. 233
Ground-state solutions of singular elliptic problems with gradient term	p. 238
Comments and historical notes	p. 241
Singular Gierer-Meinhardt systems	p. 243
Introduction	p. 243
A nonexistence result	p. 244
Existence results	p. 248
Uniqueness of the solution in one dimension	p. 255
Comments and historical notes	p. 261
Spectral theory for differential operators	p. 265
Eigenvalues and eigenfunctions for the Laplace operator	p. 265
Krein-Rutman theorem	p. 266
Implicit function theorem	p. 269
Ekeland's variational principle	p. 273
Minimization of weak lower semicontinuous coercive functionals	p. 273
Ekeland's variational principle	p. 273
Mountain pass theorem	p. 277
Ambrosetti-Rabinowitz theorem	p. 277
Application to the Emden-Fowler equation	p. 281
Mountains of zero altitude	p. 282
References	p. 283
Index	p. 295
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Singular Elliptic Problems Bifurcation & Asymptotic Analysis

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