Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45): Book

Preface vii
Chapter 1. Background Material 1
1.1 Riemannian Geometry 1
1.2 Basics in Nonlinear Analysis 7
Chapter 2. The Model Equations 13
2.1 Palais-Smale Sequences 14
2.2 Strong Solutions of Minimal Energy 17
2.3 Strong Solutions of High Energies 19
2.4 The Case of the Sphere 23
Chapter 3. Blow-up Theory in Sobolev Spaces 25
3.1 The H 2/1-Decomposition for Palais-Smale Sequences 26
3.2 Subtracting a Bubble and Nonnegative Solutions 32
3.3 The De Giorgi-Nash-Moser Iterative Scheme for Strong Solutions 45
Chapter 4. Exhaustion and Weak Pointwise Estimates 51
4.1 Weak Pointwise Estimates 52
4.2 Exhaustion of Blow-up Points 54
Chapter 5. Asymptotics When the Energy Is of Minimal Type 67
5.1 Strong Convergence and Blow-up 68
5.2 Sharp Pointwise Estimates 72
Chapter 6. Asymptotics When the Energy Is Arbitrary 83
6.1 A Fundamental Estimate: 1 88
6.2 A Fundamental Estimate: 2 143
6.3 Asymptotic Behavior 182
Appendix A. The Green's Function on Compact Manifolds 201
Appendix B. Coercivity Is a Necessary Condition 209
Bibliography 213