## Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian SystemsHilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radó. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field. The fourth edition gives a survey on new developments in the field. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Also the recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. Aside from these more significant additions, a number of smaller changes throughout the text have been made and the references have been updated. |

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### Contents

1 | |

1 Lower Semicontinuity | 2 |

2 Constraints | 13 |

3 Compensated Compactness | 25 |

4 The ConcentrationCompactness Principle | 36 |

5 Ekelands Variational Principle | 51 |

6 Duality | 58 |

7 Minimization Problems Depending on Parameters | 69 |

10 Critical Points of Mountain Pass Type | 143 |

11 Nondifferentiable Functionals | 150 |

12 LusternikSchnirelman Theory on Convex Sets | 162 |

Limit Cases of the PalaisSmale Condition | 169 |

1 Pohozaevs Nonexistence Result | 170 |

2 The BrezisNirenberg Result | 173 |

3 The Effect of Topology | 183 |

4 The Yamabe Problem | 194 |

Minimax Methods | 74 |

2 The PalaisSmale Condition | 77 |

3 A General Deformation Lemma | 81 |

4 The Minimax Principle | 87 |

5 Index Theory | 94 |

6 The Mountain Pass Lemma and its Variants | 108 |

7 Perturbation Theory | 118 |

8 Linking | 125 |

9 Parameter Dependence | 137 |