Normalized solutions for Sobolev critical Schr\"odinger equations on bounded domains

We study the existence and multiplicity of positive solutions with prescribed L²-norm for the Sobolev critical Schr\"odinger equation on a bounded domain N, N3: \ - U = U + U^2^{*-1}, U H¹₀ (), _ U²\, dx = ^2, \ where 2^*=2NN-2. First, we consider a general bounded domain in dimension N3, 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 L²-sphere, for belonging to a subset of positive measure of the interval (0, ^**), for a suitable threshold ^**>0. Next, assuming that is star-shaped, we extend the previous result to all values (0, ^**). 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.