Ricerc@Sapienza

We study weak solutions of the problem
$$
\begin{dcases*}
\ - \Delta u = \frac{\lambda}{|x|^2} u + u^p & \ \ \ in \ $\Omega \backslash\{0\}$\\
\ u \geq 0 & \ \ \ in \ $\Omega \backslash\{0\}$\\
\ u|_{\partial \Omega} =0 &
\end{dcases*}
$$
where $\Omega \subseteq \real^N$ is a smooth bounded domain containing the origin, $N \geq 3$, $1

Over the past decade, the rogue wave debate has stimulated the comparison of predictions and observations among different branches of wave physics, particularly between hydrodynamics and optics, in situations where analogous dynamical behaviors can be identified, thanks to the use of common universal models. Although the scalar nonlinear Schrödinger equation (NLSE) has constantly played a central role for rogue wave investigations, moving beyond the standard NLSE model is relevant and needful for describing more general classes of physical systems and applications.