In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In particular, we focus on the fractional Laplacian operator and on sources involving convolution with the Riesz potential, as well as on the interaction of the two, and we aim to do it through variational and topological methods.
Throughout the thesis we develop some new tools: among them, a Lagrangian formulation modeled on Pohozaev mountains is used for the existence of normalized solutions, annuli-shaped multidimensional paths are built for genus-based multiplicity results, a fractional chain rule is proved to treat concave powers, and a fractional center of mass is defined to detect semiclassical standing waves. We believe that these tools could be used to face problems in different frameworks as well.