Entire Solutions of Semilinear Elliptic Equations

ISBN-13: 9783034899628 / Angielski / Miękka / 2011 / 260 str.

Ilya A. Kuzin; Stanislav I. Pohozaev

Opis

Semilinear elliptic equations play an important role in many areas of mathematics and its applications to physics and other sciences. This book presents a wealth of modern methods to solve such equations, including the systematic use of the Pohozaev identities for the description of sharp estimates for radial solutions and the fibring method. Existence results for equations with supercritical growth and non-zero right-hand sides are given.
Readers of this exposition will be advanced students and researchers in mathematics, physics and other sciences who want to learn about specific methods to tackle problems involving semilinear elliptic equations.

Spis treści

§ 0 Notation.- 1 Classical Variational Method.- § 1 Preliminaries.- § 2 The Classical Method: Absolute Minimum.- § 3 Approximation by Bounded Domains.- § 4 Approximation for Problems on an Absolute Minimum.- § 5 The Monotonicity Method. Uniqueness of Solutions.- 2 Variational Methods for Eigenvalue Problems.- § 6 Abstract Theorems.- § 7 The Equation —?u + a(X) |u|p?2u ? ?b|u|q?2u = 0.- § 8 Radial Solutions —?u + ?f(u) = 0.- § 9 The Equation —?u ? ?|u|p?2u ? b|u|q?2u = 0.- § 10 The Equation.- § 11 The Comparison Method for Eigenvalue Problems (Concentration Compactness).- § 12 Homogeneous Problems.- 3 Special Variational Methods.- § 13 The Mountain Pass Method.- § 14 Behavior of PS-sequences. The Concentration Compactness (Comparison) Method.- § 15 A General Comparison Theorem. The Ground State. Examples for the Mountain Pass Method.- § 16 Behavior of PS-sequences in the Symmetric Case. Existence Theorems.- § 17 Nonradial Solutions of Radial Equations.- § 18 Methods of Bounded Domains Approximation.- 4 Radial Solutions: The ODE Method.- § 19 Basic Techniques of the ODE Method.- § 20 Autonomous Equations in the N-dimensional Case.- § 21 Decaying Solutions. The One-dimensional Case.- § 22 The Phase Plane Method. The Emden-Fowler Equatio.- § 23 Scaling.- § 24 Positive Solutions. The Shooting Method.- 5 Other Methods.- § 25 The Method of Upper and Lower Solutions.- § 26 The Leray-Schauder Method.- § 27 The Method of A Priori Estimates.- § 28 The Fibering Method. Existence of Infinitely Many Solutions.- § 29 Nonexistence Results.- Appendices.- A Spaces and Functionals.- B The Strauss Lemma.- C Invariant Spaces.- D The Schwarz Rearrangement.- E The Mountain Pass Method.- References.