On radial positive normalized solutions of the Nonlinear Schr\"odinger equation in an annulus

Abstract

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.