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

We are interested in the following semilinear elliptic problem: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \lambda u = u^{p-1}, x \in T, \\ u > 0, u = 0 \ \text {on} \ \partial T, \\ \int _{T}u^{2} \, dx= c \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+\lambda u=u^{p-1},x\in T,\hfill \\ u>0,u=0\text{on}\partial T,\hfill \\ {\int }_T{u}^{2}dx=c\hfill \end{array}\right)\end{array}$$ where \(T = \{x \in \mathbb {R}^{N}: 1< |x| < 2\}\) $T=\{x\in {\mathbb{R}}^N:1<|x|<2\}$ is an annulus in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ , \(N \ge 2\) $N\ge 2$ , \(p > 1\) $p>1$ 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 \(\lambda \rightarrow +\infty \) $\lambda \to +\infty$ and \(\lambda \rightarrow -\lambda _1\) $\lambda \to -{\lambda }_1$ 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 Pierotti et al. in Calc Var Partial Differ Equ 56:1–27, 2017. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained when \(N \ge 3\) $N\ge 3$ or if \(N = 2\) $N=2$ and \(p < 6\) $p<6$ . Our paper also includes the demonstration of orbital stability/instability results.