NORMALIZED SOLUTIONS FOR SOBOLEV CRITICAL SCHRODINGER EQUATIONS ON BOUNDED DOMAINS

We study the existence and multiplicity of positive solutions with prescribed L2norm for the Sobolev critical Schróodinger equation on a bounded domain \Omega \subset \BbbRN, N \geq 3,-\DeltaU = \lambdaU + U2\ast-1, U \in H01(\Omega), \int\Omega U2 dx = \rho2, where 2\ast = N2-N2 . First, we consider a general bounded domain \Omega in dimension N \geq 3, 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 \rho belonging to a subset of positive measure of the interval (0, \rho\ast\ast) and for a suitable threshold \rho\ast\ast > 0. Next, assuming that \Omega is star-shaped, we extend the previous result to all values \rho \in (0, \rho\ast\ast). 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.