Fractional Elliptic Problems with Critical Growth in the Whole of R 1st edition by Serena Dipierro, Maria Medina, Enrico Valdinoci ISBN 8876426019 9788876426018

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ISBN 10:  8876426019  
ISBN 13: 978-8876426018
Author:    Serena Dipierro, María Medina, Enrico Valdinoci

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.

Fractional Elliptic Problems with Critical Growth in the Whole of R” 1st Table of contents:

1. Introduction

1.1 The Fractional Laplacian
1.2 The Mountain Pass Theorem
1.3 The Concentration-Compactness Principle

2. The Problem Studied in This Monograph

2.1 Fractional Critical Problems
2.2 An Extended Problem and Statement of the Main Results

3. Functional Analytical Setting

3.1 Weighted Sobolev Embeddings
3.2 A Concentration-Compactness Principle

4. Existence of a Minimal Solution and Proof of Theorem 2.2.2

4.1 Some Convergence Results in View of Theorem 2.2.2
4.2 Palais-Smale Condition for Fε
4.3 Proof of Theorem 2.2.2

5. Regularity and Positivity of the Solution

5.1 A Regularity Result
5.2 A Strong Maximum Principle and Positivity of the Solutions

6. Existence of a Second Solution and Proof of Theorem 2.2.4

6.1 Existence of a Local Minimum for Iε
6.2 Some Preliminary Lemmas Towards the Proof of Theorem 2.2.4
6.3 Some Convergence Results in View of Theorem 2.2.4
6.4 Palais-Smale Condition for Iε
6.5 Bound on the Minimax Value
6.6 Proof of Theorem 2.2.4

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