Semilinear Elliptic Equations for Beginners

Marino Badiale • Enrico Serra Semilinear Elliptic Equations for Beginners Existence Results via the Variational Approach Springer . Contents 1 Introdu...

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Marino Badiale • Enrico Serra

Semilinear Elliptic Equations for Beginners Existence Results via the Variational Approach

Springer

Contents

1

Introduction and Basic Results 1.1 Motivations and Brief Historical Notes 1.2 Notation and Preliminaries 1.2.1 Function Spaces 1.2.2 Embeddings 1.2.3 Elliptic Equations 1.2.4 Frequently Used Results 1.3 A Review of Differential Calculus for Real Functionals 1.3.1 Examples in Abstract Spaces 1.3.2 Examples in Concrete Spaces 1.4 Weak Solutions and Critical Points 1.5 Convex Functionals 1.6 A Few Examples 1.7 Some Spectral Properties of Elliptic Operators 1.8 Exercises 1.9 Bibliographical Notes

1 1 5 5 7 8 9 11 15 16 22 25 28 31 35 37

2

Minimization Techniques: Compact Problems 2.1 Coercive Problems 2.2 A min-max Theorem 2.3 Superlinear Problems and Constrained Minimization 2.3.1 Minimization on Spheres 2.3.2 Minimization on the Nehari Manifold 2.4 A Perturbed Problem 2.5 Nonhomogeneous Nonlinearities 2.6 The p-Laplacian 2.6.1 Basic Theory 2.6.2 Two Applications 2.7 Exercises 2.8 Bibliographical Notes

39 39 46 55 56 59 63 74 86 87 91 93 95 ix

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Contents

3

Minimization Techniques: Lack of Compactness 3.1 A Radial Problem in RN 3.2 A Problem with Unbounded Potential 3.3 A Serious Loss of Compactness 3.4 Problems with Critical Exponent 3.4.1 The Prototype Problem 3.4.2 A Problem with a Radial Coefficient 3.4.3 A Nonexistence Result 3.5 Exercises 3.6 Bibliographical Notes

97 97 104 107 119 120 129 136 140 143

4

Introduction to Minimax Methods 4.1 Deformations 4.2 The Minimax Principle 4.3 Two Classical Theorems 4.4 Some Applications 4.4.1 Superlinear Problems 4.4.2 Asymptotically Linear Problems 4.4.3 A Problem at Resonance 4.5 Problems with a Parameter 4.6 Exercises 4.7 Bibliographical Notes

145 146 153 155 160 160 167 171 178 186 188

Index of the Main Assumptions

191

References

193

Index

197