Ricerc@Sapienza

The present research project focuses on some specific topics arising in the theory of nonlinear elliptic partial differential equations, ranging from the uniformly elliptic regime to some strongly degenerate cases.
More precisely, we plan to study the following problems:

1) Uniformly Elliptic Equations, with special attention to
-Radial critical exponents for Pucci's operators;
-Morse index for radial sign-changing solutions of classical Lane-
Emden and Hénon equations;
-Overdetermined elliptic problems;
- Critical points of Moser-Trudinger functional on surfaces.
2) Systems of elliptic equations, in particular including
- Fully nonlinear systems with Pucci's operators;
- Reaction-Diffusion Systems in population dynamics.
3) Nonlocal and degenerate elliptic equations, focusing on
- Fractional  truncated Laplacians;
- Mixed boundary problems for degenerate/singular equations;
- Inequalities for pseudo-differential operators.

The methodology we intend to implement involves different techniques, from a dynamical systems approach for radial uniformly elliptic equations to comparison principles and ad-hoc constructed barrier functions for the case of strongly degenerate operators, passing through bifurcation methods for the variational cases.
The proposed research can lead to consistent advances with respect to the current state of the art in the theory of nonlinear elliptic equations. Moreover, the topics under investigation have relevance in applied models and/or play a role in neighboring fields. The proposed research will be carried on by means of already well established collaborations of the participants with international young researchers and leading experts in the field.