Progress in Nonlinear Differential Equations and their Applications

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.

Variational Methods for Nonlocal Fractional Problems

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave—convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

/pdf/a-concave-convex-elliptic-problem-involving-the-fractional-1ynt6mzh2d.pdf

A concave—convex elliptic problem involving the fractional Laplacian

https://www.sissa.it/fa/download/seminars/ColoradoSISSA2011.pdf

On some critical problems for the fractional Laplacian operator

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.

Antonio Ambrosetti

Papers

Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems

A note on the existence of multiple brake orbits