On radial positive normalized solutions of the Nonlinear Schrödinger equation in an annulus

We are interested in the following semilinear elliptic problem: \begin{equation*} \begin{cases} -Δu + λu = u^{p-1} \ $\text{in}$ \ T,\ u > 0, u = 0 \ $\text{on}$ \ \partial T,\ \int_{T}u^{2} , dx= c \end{cases} \end{equation*} where $T=\{x\in {\mathbb{R}}^N:1<\mathrm{\mid }x\mathrm{\mid }<2\}$ is an annulus in ${\mathbb{R}}^N$, $N\ge 2$, $p>1$ is Sobolev-subcritical, searching for conditions (about $c$, $N$ and $p$) for the existence of positive radial solutions. We analyze the asymptotic behavior of $c$ as $\lambda \to +\mathrm{\infty }$ and $\lambda \to -{\lambda }_1$ to get the existence, non-existence and multiplicity of normalized solutions. Additionally, based on the properties of these solutions, we extend the results obtained in \cite{pierotti2017normalized}. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained when $N\ge 3$ or if $N=2$ and $p<6$. Our paper also includes the demonstration of orbital stability/instability results.