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Normalized bound states for the nonlinear Schrodinger equation in bounded domains

Given $ρ>0$, we study the elliptic problem \[ \text{find } (U,λ)\in H^1_0(Ω)\times \mathbb{R} \text{ such that } \begin{cases} -ΔU+λU=|U|^{p-1}U \int_Ω U^2\, dx=ρ, \end{cases} \] where $Ω\subset\mathbb{R}^N$ is a bounded domain and $p>1$ is Sobolev-subcritical, searching for conditions (about $ρ$, $N$ 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 $ρ>0$. 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 $ρ$ is sufficiently small. Next we provide existence results for certain ranges of $ρ$, which can be estimated in terms of the Dirichlet eigenvalues of $-Δ$ in $H^1_0(Ω)$, 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.