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

Abstract

We are interested in the following semilinear elliptic problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \lambda u = u^{p-1}, x \in T, \\ u > 0, u = 0 \ \text {on} \ \partial T, \\ \int _{T}u^{2} \, dx= c \end{array}\right. } \end{aligned}$$

where $T = \{x \in \mathbb {R}^{N}: 1< |x| < 2\}$ 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 $\lambda \rightarrow +\infty$ 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 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 $N \ge 3$ or if $N = 2$ and $p < 6$. Our paper also includes the demonstration of orbital stability/instability results.