Using suitable topological and variational methods it is possible to consider problems related to the existence, multiplicity and qualitative properties of solutions of several nonlinear problems involving nonlocal operators of fractional types. We recall that in recent years these operators have been extensively studied due to their application in several areas of research such as optimization, finance, phase transition, anomalous diffusions, crystal dislocation, conservation laws and quantum mechanics.
Nonlocal problems are not partial differential equations but rather integral equations. The main difficulty in the study of nonlocal problems is related to the fact that the leading operator has to take care of the behavior of solutions in the whole space and not only locally. This is in striking contrast with classical PDEs which are governed by local differential operators like the Laplacian.
In particular way, we consider heteroclinic solutions for fractional Allen-Cahn equations, positive and nodal solutions for nonlinear fractional Schrödinger equations in RN with different type of potentials and nonlinearities having subcritical, critical and supercritical growth, fractional (degenerate and not degenerate) Kirchhoff type problems, periodic solutions for fractional equations, and nontrivial complex valued solutions for nonlocal equations involving fractional operators with magnetic fields.