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

TL;DR

This paper investigates the existence and orbital stability of ground state solutions with prescribed mass for the L^2-critical and supercritical nonlinear Schrödinger equations on bounded domains, providing conditions and stability results.

Contribution

It establishes necessary and sufficient conditions for solutions' existence and analyzes their orbital stability in different nonlinear regimes.

Findings

Standing waves are orbitally stable in the L^2-critical/subcritical case.

Standing waves are almost always stable in the L^2-supercritical case.

A variational problem with two constraints is studied, which is of independent interest.

Abstract

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.