A very interesting area of nonlinear analysis lies in the study of elliptic equations involving fractional operators. Recently, great attention has been focused on these problems, both for pure mathematical research and in view of concrete real-world applications. Indeed, this type of operator arises in a quite natural way in different contexts, such as the description of several physical phenomena.

The current literature on these abstract tools and on their applications is therefore very interesting and, up to now, quite large. Motivated by this increasing interest on these subjects, this book deals with some classes of fractional problems widely investigated by many mathematicians and scientists.

This monograph is divided into three parts and is based on results obtained by ourselves or through direct cooperation with other mathematicians. More precisely, the first part deals with some basic facts about fractional Sobolev spaces, and the second part is dedicated to an analysis of fractional elliptic problems involving subcritical nonlinearities via classical variational methods and other novel approaches. Finally, in the third part of the book we give a selection of recent results on critical fractional equations, studied in the recent literature, also in relation to the celebrated Brezis–Nirenberg problem. Of course, there are many other interesting applications and theoretical aspects of fractional nonlocal problems, but it is not our ambition to treat all these topics here.

This book is addressed to researchers and advanced graduate students specializing in the fields of fractional elliptic equations, nonlinear analysis, and functional analysis. We also emphasize that the bibliography does no escape the usual rule, being incomplete. Indeed, we have listed only papers that are closer to the topics discussed in this book. But we are afraid that even for these arguments, the references are far from being exhaustive. We apologize for possible omissions.

We emphasize that this book would never have appeared without the encouragements of some dear friends and colleagues. It is a pleasure to thank some of them, especially Rossella Bartolo, Xavier Cabré, Philippe Ciarlet, Bernard Dacorogna, Alessio Fiscella, Jean Mawhin, Giuseppe Mingione, Giampiero Palatucci, Patrizia Pucci, Gaetana Restuccia, Biagio Ricceri, Dušan Repovš, Sandro Salsa, Simone Secchi, Francesco Tulone, Enrico Valdinoci, Gianmaria Verzini, and Binlin Zhang.