Singular elliptic problems: bifurcation and asymptotic analysis
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Clarendon Press [u.a.]
2008
|
| Series: | Oxford lecture series in mathematics and its applications
37 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [283]-294) and index |
| Physical Description: | XVI, 298 S. graph. Darst. |
| ISBN: | 9780195334722 0195334728 |
Staff View
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| 245 | 1 | 0 | |a Singular elliptic problems |b bifurcation and asymptotic analysis |c Marius Ghergu ; Vicenţiu D. Rădulescu |
| 264 | 1 | |a Oxford [u.a.] |b Clarendon Press [u.a.] |c 2008 | |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
I PRELIMINARIES
Basic methods
3
1.1
A fixed point result
3
1.2
The method of sub- and
supersolution
5
1.3
Comparison principles
9
1.3.1
Weak and strong maximum principle
9
1.3.2
Maximum principle for weakly differentiable functions
10
1.3.3
Stampacchia s maximum principle
11
1.3.4
Vazquez s maximum principle
12
1.3.5
Pucci
and Serrin s maximum principle
15
1.3.6
A comparison principle in the presence of singular
nonlinearities
17
1.4
Existence properties and related maximum principles
20
1.4.1
Dead core solutions of sublinear logistic equations
23
1.4.2
Singular solutions of the logistic equation
24
1.5
Brezis-Oswald theorem
26
1.6
Comments and historical notes
32
II BLOW-UP SOLUTIONS
Blow-up solutions for
semilinear
elliptic equations
37
2.1
Introduction
. 37
2.2
Blow-up solution for elliptic equations with vanishing
potential
40
2.2.1
Existence results in bounded domains
40
2.2.2
Existence results in the whole space
42
2.3
Blow-up solutions for logistic equations
45
2.3.1
The case of positive potentials
45
2.3.2
The case of vanishing potentials
46
2.4
An equivalent criterion to the Keller-Osserman condition
49
2.5
Singular solutions of the logistic equation on domains with
holes
52
2.6
Uniqueness of blow-up solution
58
2.7
A Karamata theory approach for uniqueness of blow-up
solution
64
2.8
Comments and historical notes
71
Entire solutions blowing up at infinity for elliptic systems
75
3.1
Introduction
75
xiv Contents
3.2 Characterization
of the central value set
76
3.2.1
Bounded or unbounded entire solutions
76
3.2.2
Role of the Keller-Osserman condition
81
3.3
Comments and historical notes
89
III ELLIPTIC PROBLEMS WITH SINGULAR
NONLINEARITIES
4
Sublinear perturbations of singular elliptic problems
93
4.1
Introduction
93
4.2
An ODE with mixed nonlinearities
98
4.3
A complete description for positive potentials
99
4.4
An example
104
4.5
Bifurcation for negative potentials
107
4.0
Existence for large values of parameters in the sign-changing
case
112
4.7
Singular elliptic problems in the whole space
113
4.7.1
Existence of entire solutions
113
4.7.2
Uniqueness of radially symmetric solutions
117
4.8
Comments and historical notes
121
5
Bifurcation and asymptotic analysis: The monotone case
125
5.1
Introduction
126
5.2
A general bifurcation result
127
5.3
Existence and bifurcation results
129
5.4
Asymptotic behavior of the solution with respect to
parameters
134
5.5
Examples
137
5.5.1
First example
137
5.5.2
Second example
138
5.6
The case of singular nonlinearities
140
5.7
Comments and historical notes
142
6
Bifurcation and asymptotic analysis: The
nonmonotone
case
143
6.1
Introduction
143
ß.2
Auxiliary results
144
6.3
Existence and bifurcation results in the
nonmonotone case
147
6.4
An example
154
6.5
Comments and historical notes
154
7
Superlinear perturbations of singular elliptic problems
157
7.1
Introduction
157
7.2
The weak sub- and
supersolution
method
158
7.3
Hl local tntntmizers
161
7.4
Existence of the first solution
163
Contents xv
7.5
Existence
of the second solution
169
7.5.1
First case
170
7.5.2
Second case
174
7.6
C1 regularity of solution
182
7.7
Asymptotic behavior of solutions
184
7.8
Comments and historical notes
189
8
Stability of the solution of a singular problem
191
8.1
Stability of the solution in a general singular setting
191
8.2
A min-max
characterization of the first eigenvalue for the
linearized problem
199
8.3
Differentiability of some singular nonlinear problems
201
8.4
Examples
204
8.5
Comments and historical notes
205
9
The influence of a nonlinear convection term in singular
elliptic problems
207
9.1
Introduction
207
9.2
A general nonexistence result
208
9.3
A singular elliptic problem in one dimension
210
9.4
Existence results in the
sublinear case
215
9.5
Existence results in the linear case
226
9.6
Boundary estimates of the solution
228
9.7
The case of a negative singular potential
231
9.7.1
A nonexistence result
232
9.7.2
Existence result
233
9.8
Ground-state solutions of singular elliptic problems with
gradient term
238
9.9
Comments and historical notes
241
10
Singular Gierer—Meinhardt systems
243
10.1
Introduction
243
10.2
A nonexistence result
244
10.3
Existence results
248
10.4
Uniqueness of the solution in one dimension
255
10.5
Comments and historical notes
261
Appendix A Spectral theory for differential operators
265
A.I Eigenvalues and eigenfunctions for the Laplace operator
265
A.2 Krein-Rutman theorem
266
Appendix
В
Implicit function theorem
269
Appendix
С
Ekeland s variational principle
273
C.I Minimization of weak lower semicontinuous coercive
functional
273
C.2
Ekelanďs
variational principle
273
xvi
Contents
Appendix
D
Mountain pass theorem
277
D.I Ambrosetti-Rabinowitz theorem
277
D.2 Application to the Emden-Fowler equation
281
D.3 Mountains of zero altitude
282
References
283
Index
295
|
| adam_txt |
CONTENTS
I PRELIMINARIES
Basic methods
3
1.1
A fixed point result
3
1.2
The method of sub- and
supersolution
5
1.3
Comparison principles
9
1.3.1
Weak and strong maximum principle
9
1.3.2
Maximum principle for weakly differentiable functions
10
1.3.3
Stampacchia's maximum principle
11
1.3.4
Vazquez's maximum principle
12
1.3.5
Pucci
and Serrin's maximum principle
15
1.3.6
A comparison principle in the presence of singular
nonlinearities
17
1.4
Existence properties and related maximum principles
20
1.4.1
Dead core solutions of sublinear logistic equations
23
1.4.2
Singular solutions of the logistic equation
24
1.5
Brezis-Oswald theorem
26
1.6
Comments and historical notes
32
II BLOW-UP SOLUTIONS
Blow-up solutions for
semilinear
elliptic equations
37
2.1
Introduction
. 37
2.2
Blow-up solution for elliptic equations with vanishing
potential
40
2.2.1
Existence results in bounded domains
40
2.2.2
Existence results in the whole space
42
2.3
Blow-up solutions for logistic equations
45
2.3.1
The case of positive potentials
45
2.3.2
The case of vanishing potentials
46
2.4
An equivalent criterion to the Keller-Osserman condition
49
2.5
Singular solutions of the logistic equation on domains with
holes
52
2.6
Uniqueness of blow-up solution
58
2.7
A Karamata theory approach for uniqueness of blow-up
solution
64
2.8
Comments and historical notes
71
Entire solutions blowing up at infinity for elliptic systems
75
3.1
Introduction
75
xiv Contents
3.2 Characterization
of the central value set
76
3.2.1
Bounded or unbounded entire solutions
76
3.2.2
Role of the Keller-Osserman condition
81
3.3
Comments and historical notes
89
III ELLIPTIC PROBLEMS WITH SINGULAR
NONLINEARITIES
4
Sublinear perturbations of singular elliptic problems
93
4.1
Introduction
93
4.2
An ODE with mixed nonlinearities
98
4.3
A complete description for positive potentials
99
4.4
An example
104
4.5
Bifurcation for negative potentials
107
4.0
Existence for large values of parameters in the sign-changing
case
112
4.7
Singular elliptic problems in the whole space
113
4.7.1
Existence of entire solutions
113
4.7.2
Uniqueness of radially symmetric solutions
117
4.8
Comments and historical notes
121
5
Bifurcation and asymptotic analysis: The monotone case
125
5.1
Introduction
126
5.2
A general bifurcation result
127
5.3
Existence and bifurcation results
129
5.4
Asymptotic behavior of the solution with respect to
parameters
134
5.5
Examples
137
5.5.1
First example
137
5.5.2
Second example
138
5.6
The case of singular nonlinearities
140
5.7
Comments and historical notes
142
6
Bifurcation and asymptotic analysis: The
nonmonotone
case
143
6.1
Introduction
143
ß.2
Auxiliary results
144
6.3
Existence and bifurcation results in the
nonmonotone case
147
6.4
An example
154
6.5
Comments and historical notes
154
7
Superlinear perturbations of singular elliptic problems
157
7.1
Introduction
157
7.2
The weak sub- and
supersolution
method
158
7.3
Hl local tntntmizers
161
7.4
Existence of the first solution
163
Contents xv
7.5
Existence
of the second solution
169
7.5.1
First case
170
7.5.2
Second case
174
7.6
C1 regularity of solution
182
7.7
Asymptotic behavior of solutions
184
7.8
Comments and historical notes
189
8
Stability of the solution of a singular problem
191
8.1
Stability of the solution in a general singular setting
191
8.2
A min-max
characterization of the first eigenvalue for the
linearized problem
199
8.3
Differentiability of some singular nonlinear problems
201
8.4
Examples
204
8.5
Comments and historical notes
205
9
The influence of a nonlinear convection term in singular
elliptic problems
207
9.1
Introduction
207
9.2
A general nonexistence result
208
9.3
A singular elliptic problem in one dimension
210
9.4
Existence results in the
sublinear case
215
9.5
Existence results in the linear case
226
9.6
Boundary estimates of the solution
228
9.7
The case of a negative singular potential
231
9.7.1
A nonexistence result
232
9.7.2
Existence result
233
9.8
Ground-state solutions of singular elliptic problems with
gradient term
238
9.9
Comments and historical notes
241
10
Singular Gierer—Meinhardt systems
243
10.1
Introduction
243
10.2
A nonexistence result
244
10.3
Existence results
248
10.4
Uniqueness of the solution in one dimension
255
10.5
Comments and historical notes
261
Appendix A Spectral theory for differential operators
265
A.I Eigenvalues and eigenfunctions for the Laplace operator
265
A.2 Krein-Rutman theorem
266
Appendix
В
Implicit function theorem
269
Appendix
С
Ekeland's variational principle
273
C.I Minimization of weak lower semicontinuous coercive
functional
273
C.2
Ekelanďs
variational principle
273
xvi
Contents
Appendix
D
Mountain pass theorem
277
D.I Ambrosetti-Rabinowitz theorem
277
D.2 Application to the Emden-Fowler equation
281
D.3 Mountains of zero altitude
282
References
283
Index
295 |
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| discipline | Mathematik |
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| id | DE-604.BV023367147 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:11:14Z |
| indexdate | 2024-07-09T21:16:58Z |
| institution | BVB |
| isbn | 9780195334722 0195334728 |
| language | English |
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| physical | XVI, 298 S. graph. Darst. |
| publishDate | 2008 |
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| series | Oxford lecture series in mathematics and its applications |
| series2 | Oxford lecture series in mathematics and its applications |
| spelling | Ghergu, Marius Verfasser aut Singular elliptic problems bifurcation and asymptotic analysis Marius Ghergu ; Vicenţiu D. Rădulescu Oxford [u.a.] Clarendon Press [u.a.] 2008 XVI, 298 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford lecture series in mathematics and its applications 37 Includes bibliographical references (p. [283]-294) and index Equações diferenciais larpcal Teoria assintótica larpcal Differential equations, Elliptic Asymptotic theory Differential equations, Nonlinear Bifurcation theory Singuläre partielle Differentialgleichung (DE-588)4729972-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Singuläre partielle Differentialgleichung (DE-588)4729972-1 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Rădulescu, Vicenţiu D. 1958- Verfasser (DE-588)138708924 aut Oxford lecture series in mathematics and its applications 37 (DE-604)BV009910017 37 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016550477&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ghergu, Marius Rădulescu, Vicenţiu D. 1958- Singular elliptic problems bifurcation and asymptotic analysis Oxford lecture series in mathematics and its applications Equações diferenciais larpcal Teoria assintótica larpcal Differential equations, Elliptic Asymptotic theory Differential equations, Nonlinear Bifurcation theory Singuläre partielle Differentialgleichung (DE-588)4729972-1 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4729972-1 (DE-588)4078889-1 (DE-588)4014485-9 |
| title | Singular elliptic problems bifurcation and asymptotic analysis |
| title_auth | Singular elliptic problems bifurcation and asymptotic analysis |
| title_exact_search | Singular elliptic problems bifurcation and asymptotic analysis |
| title_exact_search_txtP | Singular elliptic problems bifurcation and asymptotic analysis |
| title_full | Singular elliptic problems bifurcation and asymptotic analysis Marius Ghergu ; Vicenţiu D. Rădulescu |
| title_fullStr | Singular elliptic problems bifurcation and asymptotic analysis Marius Ghergu ; Vicenţiu D. Rădulescu |
| title_full_unstemmed | Singular elliptic problems bifurcation and asymptotic analysis Marius Ghergu ; Vicenţiu D. Rădulescu |
| title_short | Singular elliptic problems |
| title_sort | singular elliptic problems bifurcation and asymptotic analysis |
| title_sub | bifurcation and asymptotic analysis |
| topic | Equações diferenciais larpcal Teoria assintótica larpcal Differential equations, Elliptic Asymptotic theory Differential equations, Nonlinear Bifurcation theory Singuläre partielle Differentialgleichung (DE-588)4729972-1 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Equações diferenciais Teoria assintótica Differential equations, Elliptic Asymptotic theory Differential equations, Nonlinear Bifurcation theory Singuläre partielle Differentialgleichung Verzweigung Mathematik Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016550477&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009910017 |
| work_keys_str_mv | AT ghergumarius singularellipticproblemsbifurcationandasymptoticanalysis AT radulescuvicentiud singularellipticproblemsbifurcationandasymptoticanalysis |