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