Variational Methods for Nonlocal Fractional Problems Encyclopedia of Mathematics and its Applications Series

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.

Giovanni Molica Bisci is Assistant Professor of Mathematical Analysis at the Università 'Mediterranea' di Reggio Calabria. He is the author of more than 90 research papers in nonlinear analysis.
Vicentiu D. Radulescu is Distinguished Adjunct Professor at King Abdulaziz University in Jeddah, Saudi Arabia, and a professorial fellow at the 'Simion Stoilow' Mathematics Institute of the Romanian Academy. He is the author of several books and more than 200 research papers in nonlinear analysis. Since 2014 he is a Highly Cited Researcher (Thomson Reuters).
Raffaella Servadei is Associate Professor of Mathematical Analysis at the Università degli Studi di Urbino 'Carlo Bo'. She has authored more than 40 research papers in nonlinear analysis.