Normalized bound states for the nonlinear Schrodinger equation in bounded domains

TL;DR

This paper investigates the existence of normalized bound states for the nonlinear Schrödinger equation in bounded domains, establishing conditions based on the domain, nonlinearity, and prescribed mass, and extending previous results to general domains and sign-changing solutions.

Contribution

It provides new existence criteria for solutions with prescribed $L^2$ norm in bounded domains, including sign-changing solutions and general domains, extending prior positive solution results.

Findings

Solutions exist for all $ ho$ when $p$ is $L^2$-subcritical.

Existence of solutions with bounded Morse index for small $ ho$ in critical and supercritical cases.

Explicit estimates of $ ho$ ranges based on Dirichlet eigenvalues.

Abstract

Given $\rho>0$, we study the elliptic problem \[ \text{find } (U,\lambda)\in H^1_0(\Omega)\times \mathbb{R} \text{ such that } \begin{cases} -\Delta U+\lambda U=|U|^{p-1}U \int_{\Omega} U^2\, dx=\rho, \end{cases} \] where $\Omega\subset\mathbb{R}^N$ is a bounded domain and $p>1$ is Sobolev-subcritical, searching for conditions (about $\rho$, $N$ and $p$) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when $p$ is $L^2$-subcritical, i.e. $1<p\leq1+4/N$, the problem admits solution for every $\rho>0$. In the $L^2$-critical and supercritical case, i.e. when $1+4/N \leq p < 2^*-1$, we show that, for any $k\in\mathbb{N}$, the problem admits solutions having Morse index bounded above by $k$ only if $\rho$ is sufficiently small. Next we provide existence results for certain ranges of $\rho$, which can be estimated in terms of the Dirichlet eigenvalues of $-\Delta$ in $H^1_0(\Omega)$, extending to general domains and to changing sign solutions some results obtained in [Noris, Tavares, Verzini, Analysis & PDE, 2014] for positive solutions in the ball.