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

Abstract

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 $T=\{x\in {\mathbb{R}}^N:1<|x|<2\}$ 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 $\lambda \to +\infty$ 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.

arXiv admin note: text overlap with arXiv:2209.06665