Nonlocal elliptic PDEs with general nonlinearities

TL;DR

This thesis explores how nonlocal operators like the fractional Laplacian influence PDEs from physics, developing new mathematical tools to analyze solutions with general nonlinearities under minimal assumptions.

Contribution

It introduces novel methods such as a Lagrangian formulation, fractional chain rule, and fractional center of mass to study nonlocal PDEs with complex nonlinearities.

Findings

Proved multiplicity of solutions for nonlocal nonlinear problems.

Established regularity, positivity, and decay properties of ground states.

Analyzed the impact of potential topology on concentration phenomena.

Abstract

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. We examine both quantitative and qualitative aspects, proving multiplicity of solutions for nonlocal nonlinear problems with free or prescribed mass, showing regularity, positivity, symmetry and sharp asymptotic decay of ground states, and exploring the influence of the topology of a potential well in presence of concentration phenomena. On the nonlinearities we consider general assumptions which avoid monotonicity and homogeneity: this generality obstructs the use of classical variational tools and forces the implementation of new ideas. 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.