Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems

Squassina M.

Table of contents :
Preface……Page 3
1. Introduction……Page 4
2.1. Notions of non-smooth analysis……Page 6
2.2. The case of lower semi-continuous functionals……Page 9
2.3. Functionals of the calculus of variations……Page 12
3.1. Quasi-linear elliptic systems……Page 14
3.2. The concrete Palais-Smale condition……Page 16
3.3. Existence of multiple solutions for elliptic systems……Page 20
3.4. Regularity of weak solutions for elliptic systems……Page 21
3.5. Fully nonlinear scalar problems……Page 23
3.6. The concrete Palais-Smale condition……Page 24
3.7. Existence of a weak solution……Page 33
3.8. Super-linear problems with unbounded coefficients……Page 34
3.9. General setting and main results……Page 36
3.10. Verification of the key condition……Page 38
3.11. The variational setting……Page 39
3.12. A compactness result for J……Page 47
3.13. Proofs of the main Theorems……Page 51
3.14. Summability results……Page 54
4.1. Quasi-linear elliptic systems……Page 57
4.2. Symmetry perturbed functionals……Page 59
4.3. Boundedness of concrete Palais-Smale sequences……Page 63
4.5. Existence of multiple solutions……Page 65
4.6. Semi-linear systems with nonhomogeneous data……Page 68
4.7. Reduction to homogeneous boundary conditions……Page 70
4.8. The Palais-Smale condition……Page 75
4.9. Comparison of growths for min-max values……Page 77
4.10. Bolle’s method for non-symmetric problems……Page 79
4.11. Application to semi-linear elliptic systems……Page 80
4.12. The diagonal case……Page 83
5.1. Fully nonlinear elliptic equation……Page 84
5.2. The main result……Page 85
5.3. The concrete Palais-Smale condition……Page 87
5.4. Min-Max estimates……Page 92
5.5. Proof of the main result……Page 96
5.6. Fully nonlinear variational inequalities……Page 97
5.7. The main result……Page 98
5.8. The bounded Palais-Smale condition……Page 100
5.9. The Palais-Smale condition……Page 105
5.10. Min-Max estimates……Page 110
6.1. Positive entire solutions for fully nonlinear problems……Page 111
6.2. The concrete Palais-Smale condition……Page 113
6.3. Fully nonlinear problems at critical growth……Page 123
6.4. The first solution……Page 126
6.5. The concrete Palais-Smale condition……Page 130
6.6. The second solution……Page 133
6.7. One solution for a more general nonlinearity……Page 134
6.8. Existence of one nontrivial solution……Page 135
6.9. Problems with nearly critical growth……Page 139
6.10. The main results……Page 141
6.11. The weak limit……Page 143
6.12. Proof of the main results……Page 147
6.13. Mountain-pass critical values……Page 150
7. The Singularly Perturbed Case, I……Page 151
7.1. The del Pino-Felmer penalization scheme……Page 154
7.2. Energy estimates and concentration……Page 158
7.3. Proof of the main result……Page 168
7.4. A few related open problems……Page 171
8. The Singularly Perturbed Case, II……Page 172
8.1. Penalization and compactness……Page 174
8.2. Two consequences of the Pucci-Serrin identity……Page 179
8.3. Energy estimates……Page 184
8.4. Proofs of the main results……Page 193
9.1. A general Pucci-Serrin type identity……Page 195
9.2. The approximation argument……Page 196
9.3. Non-strict convexity in some particular cases……Page 200
9.4. The splitting case……Page 201
9.5. The one-dimensional case……Page 203
9.6. Non-existence results……Page 205
References……Page 208