Using appropriate topological and variational methods, it is possible to consider questions related to the existence, multiplicity, and qualitative properties of solutions to various nonlinear problems involving nonlocal fractional operators. In recent years, these operators have attracted considerable interest due to their application in various research fields such as optimization, finance, phase transitions, anomalous diffusion, crystal dislocations, conservation laws, quantum mechanics, and nonlocal elasticity phenomena. Nonlocal problems are not governed by partial differential equations but rather by integro-differential equations. The major difficulty in studying these is related to the fact that the carrier operator must account for the behavior of the solutions across the entire space, not just locally. This is in stark contrast to classical PDEs, which are governed by local differential operators such as the Laplacian. In particular, we consider heteroclinic solutions for fractional Allen-Cahn equations, positive and nodal solutions for fractional Schrödinger-type nonlinear equations in RN with different types of potential and nonlinearity with subcritical, critical and supercritical growth, fractional Kirchhoff-type problems (degenerate and nondegenerate), periodic solutions for fractional equations, and nontrivial complex-valued solutions for nonlocal equations with fractional operators with magnetic fields, mathematical models for the characterization of the elastic behavior of composite materials.
Prof. Vincenzo Ambrosio
email: v.ambrosio@univpm.it
