  We study the existence and multiplicity of positive solutions with prescribed
$L^2$-norm for the Sobolev critical Schr\"odinger equation on a bounded domain
$\Omega\subset\mathbb{R}^N$, $N\ge3$: \[ -\Delta U = \lambda U +
U^{2^{*}-1},\qquad U\in H^1_0(\Omega),\qquad \int_\Omega U^2\,dx = \rho^{2}, \]
where $2^*=\frac{2N}{N-2}$.
  First, we consider a general bounded domain $\Omega$ in dimension $N\ge3$,
with a restriction, only in dimension $N=3$, involving its inradius and first
Dirichlet eigenvalue. In this general case we show the existence of a mountain
pass solution on the $L^2$-sphere, for $\rho$ belonging to a subset of positive
measure of the interval $(0,\rho^{**})$, for a suitable threshold
$\rho^{**}>0$. Next, assuming that $\Omega$ is star-shaped, we extend the
previous result to all values $\rho\in(0,\rho^{**})$.
  With respect to that of local minimizers, already known in the literature,
the existence of mountain pass solutions in the Sobolev critical case is much
more elusive. In particular, our proofs are based on the sharp analysis of the
bounded Palais-Smale sequences, provided by a nonstandard adaptation of the
Struwe monotonicity trick, that we develop.
