Title and Abstract contributed talks

Rosario Corso

Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

Abstract : We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted p-Laplace operator, which we consider for a general 1<p<d where d is the dimension of the space. For p=2, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized by Dolbeault-Esteban-Loss (2016). Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

Laura Gambera

On a uniqueness result for singular elliptic problems in ℝ^N

Abstract: In this talk, concerning a work in progress, we discuss an existence and uniqueness result for singular elliptic variational systems in the whole space and driven by a non-homogeneous operator, as the (p,q)-Laplacian. The existence is obtained by solving some regularized problems through variational methods, while the uniqueness follows from a technique basically introduced by Diaz and Saa.

Umberto Guarnotta

Strongly singular convective elliptic equations in ℝ^N driven by a non-homogeneous operator

Abstract : We present an existence result for a quasi-linear elliptic problem in the whole space ℝ^N. The principal part is a non-homogeneous operator and the reaction possesses both strongly singular and convection terms.

From a technical point of view, several challenges arise in this framework:

(i) non-homogeneity of the differential operator prevents to exploit standard procedures in the construction of sub-solutions, and regularity issues do not allow to work directly in the whole space;

(ii) due to the strongly singular nature of the problem, gaining compactness from energy estimates requires some efforts, such as localization procedures and fine energy estimates on suitable super-level sets of solutions;

(iii) the loss of variational structure compels to use topological and monotonicity methods instead of variational ones;

(iv) the setting causes lack of compactness for Sobolev embeddings.

A variety of techniques will be used to ensure the existence of a generalized solution: in addition to the aforementioned tools, also regularization and approximation procedures (such as the shifting method), sub-super-solution technique, fixed point and regularity theory, maximum and comparison principles, as well as some ad hoc compactness results, are employed.