| Preface 1. Krein-Rutman Theorem and the Principal Eigenvalue 2. Maximum Principles Revisited 2.1 Equivalent forms of the maximum principle 2.2 Maximum principle in w2'N 3. The Moving Plane Method 3.1 Symmetry over bounded domains 3.2 Symmetry over the entire space 3.3 Positivity of nonnegative solutions 4. The Method of Upper and Lower Solutions 4.1 Classical upper and lower solutions 4.2 Weak upper and lower solutions 5. The Logistic Equation 5.1 The classical case 5.2 The degenerate logistic equation 5.3 Perturbation and profile of solutions 6. Boundary Blow-Up Problems 6.1 The Keller-Osserman result and its generalizations 6.2 Blow-up rate and uniqueness 6.3 Logistic type equations with weights Maximum Principles and Applications 7. Symmetry and Liouville Type Results over Half and Entire Spaces 7.1 Symmetry in a half space without strong maximum principle 7.2 Uniqueness results of logistic type equations over RN 7.3 Partial symmetry in the entire space 7.4 Some Liouville type results Appendix A Basic Theory of Elliptic Equations A.1 Schauder theory for elliptic equations A.2 Sobolev spaces A.3 Weak solutions of elliptic equations A.4 Lp theory of elliptic equations A.5 Maximum principles for elliptic equations A.5.1 The classical maximum principles A.5.2 Maximum principles and Harnack inequality for weak solutions A.5.3 Maximum principles and Harnack inequality for strong solutions Bibliography Index |
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