Existence and orbital stability of the ground states with prescribed mass for the L2-critical and supercritical NLS on bounded domains

Abstract

Given ρ > 0, we study the elliptic problem where B1 ⊂ ℝN is the unitary ball and p is Sobolev-subcritical. Such a problem arises in the search for solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about ρ, N and p) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every ρ (in the existence range) when p is L2-critical and subcritical, i.e., 1 < p < 2* - 1. The proofs are obtained in connection with the study of a variational problem with two constraints of independent interest: to maximize the Lp+1-norm among functions having prescribed L2- and H10-norms.