SEMINAR #1: On some embedding results for weighted spaces, with applications to elliptic equations (Simone Secchi)
We propose some continuous and compact embedding results for weighted Sobolev spaces whose weights are coercive potentials. These spaces are the natural setting for studying certain types of elliptic equations with variational structure.
SEMINAR #2: Variational Methods in Finsler Geometry (Giovanni Molica Bisci)
The theory of Sobolev spaces on complete Riemannian manifolds is well-established and has been extensively employed in the analysis of various elliptic problems. Although Finsler geometry constitutes a natural generalization of Riemannian geometry, the corresponding theory of Sobolev spaces on non-compact Finsler manifolds remains largely undeveloped. Motivated by the growing interest in this area within the mathematical literature, the primary aim of this talk is to present some recent results concerning non-compact Randers spaces and their applications to quasilinear elliptic equations. The approach is principally based on new abstract Sobolev embedding theorems, along with variational and topological methods developed in the recent monograph Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications, Volume 36 (2021), co-authored with P. Pucci.
SEMINAR #3: Normalized solutions of Schrödinder equations on domains (Thomas Bartsch)
The existence of solutions of nonlinear Schrödinger equations with prescribed L^2-norm has found considerable interest in the last decade.
In the entire Euclidean space and when the potential is constant the scaling $s * u(x) = s(N/2)u(sx)$ plays an important role in proving the Palais-Smale condition for the associated functional on the L^2-sphere.
We present recent results on the existence of solutions of nonlinear Schrödinger equations with prescribed L^2-norm on bounded domains, where this scaling cannot be used.
The talk is based on work with Shijie Qi and Wenming Zou.