  We study the existence of positive solutions with prescribed $L^2$-norm for
the Schr\"odinger equation
  \[ -\Delta u-V(x)u+\lambda u=|u|^{p-2}u\qquad\lambda\in \mathbb{R},\quad u\in
H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in\left(2+\frac 4
N,2^*\right)$, $2^*:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^*:=+\infty$ if $N=1,2$.
We treat two cases. Firstly, under an explicit smallness assumption on $V$ and
no condition on the mass, we prove the existence of a mountain pass solution at
positive energy level, and we exclude the existence of solutions with negative
energy. Secondly, requiring that the mass is smaller than some explicit bound,
depending on $V$, and that $V$ is not too small in a suitable sense, we find
two solutions: a local minimizer with negative energy, and a mountain pass
solution with positive energy. Moreover, a nonexistence result is proved.
