1 Introduction

1.1 Basic ideas of variational methods

1.2 Classical solution and generalized solution

Preface

1.3 First variation,Euler-Lagrange equation

1.4 Second variation

1.5 Systems

2 Sobolev Spaces

2.1 H？lder spaces

2.2 Lp spaces

2.2.1 Useful inequalities

2.2.2 Completeness of Lp(Ω)

2.2.3 Dual space of Lp(Ω)

2.2.4 Topologies in Lp(Ω)space

2.2.5 Convolution

2.2.6 Mollifier

2.3 Sobolev spaces

2.3.1 Weak derivatives

2.3.2 Definition of Sobolev spaces

2.3.3 Inequalities

2.3.4 Embedding theorems and trace theorems

3 Calculus in Banach Spaces

3.1 Frechet-derivatives

3.2 Nemyski operator

3.3 Gateaux-derivatives

3.4 Calculus of abstract functions

3.5 Initial value problem in Banach space

4 Direct Methods

4.1 Lower semi-continuity

4.2 A general lower semi-continuity result

4.3 Ekeland variational principle

4.4 Palais-Smale condition

4.5 Constrained variational problems

4.5.1 Lagrange multiplier method

4.5.2 Weak sub-and super-solutions

4.5.3 Nehari manifold

5 Deformation Theorems

5.1 Deformations in Hilbert space

5.2 Pseudo-gradient vector field

5.3 Deformations in Banach space

6.1 General minimax principle

6 Minimax Methods

6.2 Mountain pass lemma

6.3 Z2 index theory

6.3.1 Minimax principles for even functionals

6.3.2 Symmetric mountain pass lemma

6.4 Linking argument

6.5 p-Laplacian with indefinite weights

6.5.1 Linking results

6.5.2 Existence results

7.1 Pohozaev type identities

7 Noncompact Variational Problems

7.2 Symmetry and compactness

7.3 Concentration compactness principles

7.3.1 The locally compact case

7.3.2 The limit case

7.4 Unconstrained problems involving critical Sobolev exponent

7.4.1 Local compactness

7.4.2 Nonhomogeneous problem

7.4.3 Global compactness

8.1 Solitary waves

8 Generalized K-P Equation

8.1.1 Existence of nontrivial solitary waves

8.1.2 Nonexistence of nontrivial solitary waves

8.2 Stationary solutions to GKP equation in bounded domain

8.2.1 Existence results

8.2.2 Nonexistence results

9 Best Constants in Sobolev Inequalities

9.1 Best constants

9.1.1 l-dimensional case,Bliss Lemma

9.1.2 m-dimensional case,symmetrization

9.2.1 Yamabe problem

9.2 Applications

9.2.2 Problems with critical Sobolev exponents

9.2.3 Global compactness

9.3 Extensions

9.3.1 Gagliardo-Nirenberg inequality

9.3.2 The Caffarelli-Kohn-Nirenberg inequalities

Appendix A Elliptic Regularity

A.1 Local boundedness

A.2 H？lder continuity

Bibliography

Index