Fractional Elliptic Problems with Critical Growth in the Whole of $\R^n$ (Publications of the Scuola Normale Superiore Book 15)

These lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary. The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations. Front cover 1 Half title 2 Title page 4 Copyright page 5 Contents 6 Preface 8 1 Introduction 10 1.1 The fractional Laplacian 10 1.2 Themountain pass theorem 15 1.3 The concentration-compactness principle 21 2 The problem studied in this monograph 23 2.1 Fractional critical problems 23 2.2 An extended problem and statement of the main results 28 3 Functional analytical setting 36 3.1 Weighted Sobolev embeddings 36 3.2 A concentration-compactness principle 42 4 Existence of a minimal solution and proof of Theorem 2.2.2 46 4.1 Some convergence results in view of Theorem 2.2.2 46 4.2 Palais-Smale condition for Fε 49 4.3 Proof of Theorem 2.2.2 72 5 Regularity and positivity of the solution 74 5.1 Aregularity result 74 5.2 A strong maximum principle and positivity of the solutions 79 6 Existence of a second solution and proof of Theorem 2.2.4 81 6.1 Existence of a local minimum for Iε 83 6.2 Some preliminary lemmata towards the proof of Theorem 2.2.4 84 6.3 Some convergence results in view of Theorem 2.2.4 90 6.4 Palais-Smale condition for Iε 96 6.5 Bound on the minimax value 139 6.6 Proof of Theorem 2.2.4 154 References 155 Front Matter....Pages i-viii Introduction....Pages 1-13 The problem studied in this monograph....Pages 15-27 Functional analytical setting....Pages 29-38 Existence of a minimal solution and proof of Theorem 2.2.2....Pages 39-66 Regularity and positivity of the solution....Pages 67-73 Existence of a second solution and proof of Theorem 2.2.4....Pages 75-148 Back Matter....Pages 149-158