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We study solutions of a semilinear elliptic equation with prescribed mass andDirichlet homogeneous boundary conditions in the unitary ball. Such problemarises in the search of solitary wave solutions for nonlinear Schr\"odingerequations (NLS) with Sobolev subcritical power nonlinearity on bounded domains.Necessary and sufficient conditions are provided for the existence of suchsolutions. Moreover, we show that standing waves associated to least energysolutions are always orbitally stable when the nonlinearity is L^2-critical andsubcritical, while they are almost always stable in the L^2-supercriticalregime. The proofs are obtained in connection with the study of a variationalproblem with two constraints, of independent interest: to maximize theL^{p+1}-norm among functions having prescribed L^2 and H^1_0-norm.

Existence and orbital stability of the ground states with prescribed mass for the L^2-critical and supercritical NLS on bounded domains