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The paper deals with the existence of positive solutions with prescribed $L^2$
norm for the Schrödinger equation$$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$where $\Omega =\mathbb {R}^N$
or $\mathbb {R}^N\setminus \Omega$
is a compact set, $\rho >0$
, $V\ge 0$
(also $V\equiv 0$
is allowed), $p\in (2,2+\frac 4 N)$
. The existence of a positive solution $\bar u$
is proved when $V$
verifies a suitable decay assumption (Dρ), or if $\|V\|_{L^q}$
is small, for some $q\ge \frac N2$
($q>1$
if $N=2$
). No smallness assumption on $V$
is required if the decay assumption (Dρ) is fulfilled. There are no assumptions on the size of $\mathbb {R}^N\setminus \Omega$
. The solution $\bar u$
is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$
and $\Omega =\mathbb {R}^N$
.
In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality:
for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of 
$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$
${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.
