Existence And Symmetry Of Solutions For Critical Fractional Schrodinger Equations With Bounded Potentials
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| Author | : Vincenzo Ambrosio |
| Publisher | : Springer Nature |
| Total Pages | : 669 |
| Release | : 2021-04-19 |
| Genre | : Mathematics |
| ISBN | : 3030602206 |
This monograph presents recent results concerning nonlinear fractional elliptic problems in the whole space. More precisely, it investigates the existence, multiplicity and qualitative properties of solutions for fractional Schrödinger equations by applying suitable variational and topological methods. The book is mainly intended for researchers in pure and applied mathematics, physics, mechanics, and engineering. However, the material will also be useful for students in higher semesters and young researchers, as well as experienced specialists working in the field of nonlocal PDEs. This is the first book to approach fractional nonlinear Schrödinger equations by applying variational and topological methods.
| Author | : Nick Laskin |
| Publisher | : World Scientific |
| Total Pages | : 358 |
| Release | : 2018-05-28 |
| Genre | : Science |
| ISBN | : 9813223812 |
Fractional quantum mechanics is a recently emerged and rapidly developing field of quantum physics.This is the first monograph on fundamentals and physical applications of fractional quantum mechanics, written by its founder.The fractional Schrödinger equation and the fractional path integral are new fundamental physical concepts introduced and elaborated in the book. The fractional Schrödinger equation is a manifestation of fractional quantum mechanics. The fractional path integral is a new mathematical tool based on integration over Lévy flights. The fractional path integral method enhances the well-known Feynman path integral framework.Related topics covered in the text include time fractional quantum mechanics, fractional statistical mechanics, fractional classical mechanics and the α-stable Lévy random process.The book is well-suited for theorists, pure and applied mathematicians, solid-state physicists, chemists, and others working with the Schrödinger equation, the path integral technique and applications of fractional calculus in various research areas. It is useful to skilled researchers as well as to graduate students looking for new ideas and advanced approaches.
| Author | : Boling Guo |
| Publisher | : World Scientific |
| Total Pages | : 347 |
| Release | : 2015-03-09 |
| Genre | : Mathematics |
| ISBN | : 9814667064 |
This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors' beautiful hope.This book describes theoretical and numerical aspects of the fractional partial differential equations, including the authors' researches in this field, such as the fractional Nonlinear Schrödinger equations, fractional Landau-Lifshitz equations and fractional Ginzburg-Landau equations. It also covers enough fundamental knowledge on the fractional derivatives and fractional integrals, and enough background of the fractional PDEs.
| Author | : Wenxiong Chen |
| Publisher | : World Scientific |
| Total Pages | : 342 |
| Release | : 2020-06-09 |
| Genre | : Mathematics |
| ISBN | : 9813224010 |
This is a unique book that provides a comprehensive understanding of nonlinear equations involving the fractional Laplacian as well as other nonlocal operators. Beginning from the definition of fractional Laplacian, it gradually leads the readers to the frontier of current research in this area. The explanations and illustrations are elementary enough so that first year graduate students can follow easily, while it is advanced enough to include many new ideas, methods, and results that appeared recently in research literature, which researchers would find helpful. It focuses on introducing direct methods on the nonlocal problems without going through extensions, such as the direct methods of moving planes, direct method of moving spheres, direct blowing up and rescaling arguments, and so on. Different from most other books, it emphasizes on illuminating the ideas behind the formal concepts and proofs, so that readers can quickly grasp the essence.
| Author | : Giovanni Molica Bisci |
| Publisher | : Cambridge University Press |
| Total Pages | : 401 |
| Release | : 2016-03-11 |
| Genre | : Mathematics |
| ISBN | : 1316571696 |
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
| Author | : Giovanni Molica Bisci |
| Publisher | : Cambridge University Press |
| Total Pages | : 401 |
| Release | : 2016-03-11 |
| Genre | : Mathematics |
| ISBN | : 1107111943 |
A thorough graduate-level introduction to the variational analysis of nonlinear problems described by nonlocal operators.
| Author | : Nikolaos S. Papageorgiou |
| Publisher | : Springer |
| Total Pages | : 586 |
| Release | : 2019-02-26 |
| Genre | : Mathematics |
| ISBN | : 3030034305 |
This book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.
| Author | : Paul H. Rabinowitz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 110 |
| Release | : 1986-07-01 |
| Genre | : Mathematics |
| ISBN | : 0821807153 |
The book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory. The book requires some background in functional analysis and differential equations, especially elliptic partial differential equations. It is addressed to mathematicians interested in differential equations and/or nonlinear functional analysis, particularly critical point theory.
| Author | : Claudia Bucur |
| Publisher | : Springer |
| Total Pages | : 165 |
| Release | : 2016-04-08 |
| Genre | : Mathematics |
| ISBN | : 3319287397 |
Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.
| Author | : Michel Willem |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 168 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461241464 |
Many boundary value problems are equivalent to Au=O (1) where A : X --+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional rand inf.
