Variational Methods for Nonlocal Fractional Problems

A thorough graduate-level introduction to the variational analysis of nonlinear problems described by nonlocal operators.

Giovanni Molica Bisci (Author), Vicentiu D. Radulescu (Author), Raffaella Servadei (Author)

9781107111943, Cambridge University Press

Hardback, published 11 March 2016

400 pages
24 x 16.3 x 3.1 cm, 0.79 kg

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.

Foreword Jean Mawhin
Preface
Part I. Fractional Sobolev Spaces: 1. Fractional framework
2. A density result for fractional Sobolev spaces
3. An eigenvalue problem
4. Weak and viscosity solutions
5. Spectral fractional Laplacian problems
Part II. Nonlocal Subcritical Problems: 6. Mountain Pass and linking results
7. Existence and localization of solutions
8. Resonant fractional equations
9. A pseudo-index approach to nonlocal problems
10. Multiple solutions for parametric equations
11. Infinitely many solutions
12. Fractional Kirchhoff-type problems
13. On fractional Schrödinger equations
Part III. Nonlocal Critical Problems: 14. The Brezis–Nirenberg result for the fractional Laplacian
15. Generalizations of the Brezis–Nirenberg result
16. The Brezis–Nirenberg result in low dimension
17. The critical equation in the resonant case
18. The Brezis–Nirenberg result for a general nonlocal equation
19. Existence of multiple solutions
20. Nonlocal critical equations with concave-convex nonlinearities
References
Index.

Subject Areas: Mathematical physics [PHU], Topology [PBP], Calculus of variations [PBKQ], Differential calculus & equations [PBKJ], Functional analysis & transforms [PBKF]