Variational Methods for Nonlocal Fractional Problems

Abstract

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.

Multiplicity results for the non-homogeneous fractionalp-Kirchhoff equations with concave–convex nonlinearities

“…For a short introduction to the fractional Laplacian and the fractional Sobolev spaces, the reader is referred to [25]. For a detailed treatment of non-local fractional problems, we refer to the monograph [26].…”

A Global multiplicity result for a very singular critical nonlocal equation

“…The theory of fractional Laplacian and elliptic equations involving it as the principal part has been evolved immensely in recent years. There is a vast literature available on it, however we cite [6,14] for motivation to readers. The fractional elliptic equations with singular and critical nonlinearities was first studied by Barrios et al in [4].…”

An Ambrosetti–Prodi type result for fractional spectral problems

“…The literature on fractional and nonlocal operators and on their applications is so huge that we do not even try to collect here a detailed bibliography. Anyway, we refer the interested reader to for an elementary introduction on this subject.…”