1 Introduction 1
1.1 Basic ideas of variational methods 1
1.2 Classical solution and generalized solution 4
Preface 5
1.3 First variation,Euler-Lagrange equation 6
1.4 Second variation 14
1.5 Systems 14
2 Sobolev Spaces 21
2.1 H？lder spaces 21
2.2 Lp spaces 23
2.2.1 Useful inequalities 25
2.2.2 Completeness of Lp(Ω) 28
2.2.3 Dual space of Lp(Ω) 29
2.2.4 Topologies in Lp(Ω)space 31
2.2.5 Convolution 36
2.2.6 Mollifier 41
2.3 Sobolev spaces 43
2.3.1 Weak derivatives 43
2.3.2 Definition of Sobolev spaces 45
2.3.3 Inequalities 50
2.3.4 Embedding theorems and trace theorems 55
3 Calculus in Banach Spaces 63
3.1 Frechet-derivatives 63
3.2 Nemyski operator 67
3.3 Gateaux-derivatives 70
3.4 Calculus of abstract functions 74
3.5 Initial value problem in Banach space 76
4 Direct Methods 78
4.1 Lower semi-continuity 78
4.2 A general lower semi-continuity result 83
4.3 Ekeland variational principle 87
4.4 Palais-Smale condition 90
4.5 Constrained variational problems 100
4.5.1 Lagrange multiplier method 100
4.5.2 Weak sub-and super-solutions 104
4.5.3 Nehari manifold 107
5 Deformation Theorems 112
5.1 Deformations in Hilbert space 112
5.2 Pseudo-gradient vector field 118
5.3 Deformations in Banach space 122
6.1 General minimax principle 135
6 Minimax Methods 135
6.2 Mountain pass lemma 139
6.3 Z2 index theory 145
6.3.1 Minimax principles for even functionals 148
6.3.2 Symmetric mountain pass lemma 150
6.4 Linking argument 157
6.5 p-Laplacian with indefinite weights 162
6.5.1 Linking results 163
6.5.2 Existence results 165
7.1 Pohozaev type identities 169
7 Noncompact Variational Problems 169
7.2 Symmetry and compactness 174
7.3 Concentration compactness principles 181
7.3.1 The locally compact case 181
7.3.2 The limit case 191
7.4 Unconstrained problems involving critical Sobolev exponent 197
7.4.1 Local compactness 197
7.4.2 Nonhomogeneous problem 208
7.4.3 Global compactness 216
8.1 Solitary waves 224
8 Generalized K-P Equation 224
8.1.1 Existence of nontrivial solitary waves 228
8.1.2 Nonexistence of nontrivial solitary waves 231
8.2 Stationary solutions to GKP equation in bounded domain 236
8.2.1 Existence results 238
8.2.2 Nonexistence results 240
9 Best Constants in Sobolev Inequalities 245
9.1 Best constants 245
9.1.1 l-dimensional case,Bliss Lemma 246
9.1.2 m-dimensional case,symmetrization 252
9.2.1 Yamabe problem 254
9.2 Applications 254
9.2.2 Problems with critical Sobolev exponents 256
9.2.3 Global compactness 259
9.3 Extensions 260
9.3.1 Gagliardo-Nirenberg inequality 260
9.3.2 The Caffarelli-Kohn-Nirenberg inequalities 262
Appendix A Elliptic Regularity 272
A.1 Local boundedness 273
A.2 H？lder continuity 283
Bibliography 297
Index 305