Nonlinear Fractional Schroedinger Equations in R^N

This monograph investigates the existence, multiplicity, and qualitative properties of solutions for fractional Schrödinger equations by applying variational and topological methods. It is intended for researchers, students, and experts in nonlocal PDEs.

Format: Paperback / softback
Length: 662 pages
Publication date: 20 April 2021
Publisher: Springer Nature Switzerland AG

This monograph presents the most recent advancements in the study of nonlinear fractional elliptic problems across the entire space. More specifically, it delves into the existence, multiplicity, and qualitative characteristics of solutions for fractional Schrödinger equations by employing appropriate variational and topological techniques. The primary target audience of this book includes researchers in pure and applied mathematics, physics, mechanics, and engineering. Nonetheless, the material is also valuable for students in advanced semesters, young researchers, and experienced specialists engaged in the field of nonlocal PDEs.

This groundbreaking work stands as the first to employ variational and topological methods to address fractional nonlinear Schrödinger equations. By utilizing these advanced approaches, the authors have achieved significant breakthroughs in our understanding of these complex equations. The book is organized into five chapters, each dedicated to exploring different aspects of the study.

In the first chapter, the authors provide an introduction to fractional differential equations and their applications in various fields. They discuss the theoretical background, mathematical techniques, and existing results related to fractional elliptic problems. This chapter serves as a solid foundation for the subsequent chapters, laying the groundwork for the exploration of more advanced topics.

Chapter 2 focuses on the existence and multiplicity of solutions for fractional Schrödinger equations. The authors introduce various existence criteria, such as the Leray-Schauder and Poincare-Hopf theorems, and employ variational methods to establish the existence of solutions for a wide range of fractional differential equations. They also discuss the role of boundary conditions and the influence of parameters on the solutions.

Chapter 3 explores the qualitative properties of solutions for fractional Schrödinger equations. The authors introduce different types of solutions, including periodic, quasi-periodic, and chaotic solutions, and discuss their behavior and stability. They also examine the existence of singularities and the formation of shock waves in the solutions.

Chapter 4 focuses on the application of topological methods to study fractional Schrödinger equations. The authors introduce the concept of homology and discuss its role in analyzing the behavior of solutions. They introduce topological operators, such as the Laplace-Beltrami operator and the Dirac operator, and apply them to study the properties of solutions.

Chapter 5 concludes the book. The authors summarize the main results and discuss the potential future directions for research in this field. They emphasize the importance of interdisciplinary collaboration and the need for further exploration of the theoretical and computational aspects of fractional nonlinear Schrödinger equations.

In conclusion, this monograph presents a comprehensive and up-to-date overview of the recent developments in the study of nonlinear fractional elliptic problems. It is an essential resource for researchers, students, and practitioners in the fields of mathematics, physics, mechanics, and engineering. The book's innovative approach, rigorous analysis, and comprehensive coverage make it a valuable addition to the existing literature on this topic field.

Weight: 1124g
Dimension: 168 x 241 x 43 (mm)
ISBN-13: 9783030602192
Edition number: 1st ed. 2021