Rafael López Soriano (Universidad Carlos III de Madrid).

Título: Systems of elliptic equations with double criticality.
Abstract: In this talk we shall study a kind of nonlinear elliptic systems in $mathbb{R}^N$. The interest of this problem is based on the presence of critical power nonlinearities and a nonlinear coupling, possibly critical, as well as Hardy-type singular potentials. Among others, this problem was extensively studied by Abdellaoui-Felli-Peral. By means of variational methods, we will focus on the existence and compactness of solutions. More precisely, we obtain new results on ground states solutions and mountain-pass critical points of the underlying energy functional. This work is in collaboration with Eduardo Colorado and Alejandro Ortega (UC3M).

Francesca Colasuonno (Università di Bologna).

Título: Positive oscillating solutions for some Neumann problems.
Abstract: In this seminar, I will talk about some semilinear or quasilinear problems under Neumann boundary conditions. In the first part of the seminar, I will focus on supercritical p-Laplacian problems, set in a ball of R^N, and describe some existence results obtained via variational methods. In the second part, I will present some multiplicity results obtained via the shooting method for ODEs. In particular, I will describe a recent result on a one-dimensional mean-curvature problem, for which we find several positive oscillating BV-solutions. The last result is contained in a joint paper with Alberto Boscaggin and Colette De Coster.