  We are interested in the following semilinear elliptic problem:
\begin{equation*} \begin{cases} -\Delta u + \lambda u = 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 $T = \{x \in \mathbb{R}^{N}: 1 < |x| < 2\}$
is an annulus in $\mathbb{R}^{N}$, $N \geq 2$, $p > 1$ 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 $\lambda
\rightarrow +\infty$ and $\lambda \rightarrow -\lambda_1$ 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 $N \geq 3$ or
if $N = 2$ and $p < 6$. Our paper also includes the demonstration of orbital
stability/instability results.
