Normalized bound states for the nonlinear Schrodinger equation in bounded domains

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.