In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the stationary nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this talk, we consider the problem in bounded domains, in the presence of weakly attractive potentials, or under trapping potentials, and investigate the following questions:

1. whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and
2. if the existence of a mountain-pass solution persists.

We provide positive answers under suitable assumptions. This is based on joint work with Dario Pierotti and Gianmaria Verzini (Politecnico di Milano).

Let $p$ be prime. In the representation theory of $p$-adic groups over algebraically closed fields of characteristic $p$, such as the general linear group $GL_n(F)$ or the special linear group $SL_n(F)$, where the entries vary in a finite extension $F$ of the field $\mathbb{Q}_p$ of $p$-adic numbers, an important concept is that of distinguished representations. Given a subgroup $H$ of $G$ and a character $\chi$ of $H$ (i.e. a group homomorphism from $H$ to $\mathbb{K}^*$), a representation $(\pi, V)$ of $G$ is said to be $(H, \chi)$-distinguished when

\[ Hom_H (\pi, \chi) := \{l: \, V \to \chi \, \text{linear map} : \, \forall h \in H, \, v \in V, l(\pi(h)v) = \chi(h)l(v) \} \ne 0.\]

When $\chi$ is trivial (i.e. $\chi= \mathbb{1}$), we simply say that $(\pi, V)$ is $H$-distinguished. We study this concept of $p$-modular representation of $SL_2(\mathbb{F}_q)$, focusing in this talk on distinction with respect to its standard Borel subgroup of upper-triangular matrices.

This is joint work with Ramla Abdellatif (Université de Picardie Jules Verne) and Jocelyn P. Vilela (Mindanao State University-IIT).

In this seminar, I will discuss the local well-posedness (LWP) of the initial value problem for the Hirota-Satsuma system - a coupled system of two KdV-type equations. Physically, this system models nonlinear waves interactions of long waves with different dispersion relations, with applications in shallow water and stratified fluids. Since there are two equations in this system, we can suppose the initial data have different regularities, i.e., $(u_0, v_0) \in H^k (\mathbb{R}) \times H^s(\mathbb{R})$. Previous LWP results in the literature only suppose $k= s$ or $s = k + 1$. Using frequency-restricted estimates and the concept of integrated-by-parts strong solution, we were able to prove LWP in the general case $H^k (\mathbb{R}) \times H^s(\mathbb{R})$. This LWP is sharp except for only one obstruction which we are still investigating.

This is a joint work with Simão Correia and Jorge Drumond.