Normalized solutions for Sobolev critical Schr\"{o}dinger equations on bounded domains

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.