where
is the
unitary ball and
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
,
and
) are
provided for the existence of solutions. Moreover, we show that standing
waves associated to least energy solutions are orbitally stable for every
(in the existence
range) when
is
-critical and subcritical,
i.e.,
, while they are
stable for almost every
in the
-supercritical
regime
.
The proofs are obtained in connection with the study of a variational
problem with two constraints of independent interest: to maximize the
-norm among functions
having prescribed
-
and
-norms.
INdAM-COFUND Marie Curie
Fellow
Laboratoire de Mathématiques
Université de Versailles Saint-Quentin-en-Yvelines
45 avenue des Étas-Unis
78035 Versailles
France
Universidade de Lisboa
Centro de Matemática e Aplicações Fundamentais and Faculdade de
Ciências da Universidade de Lisboa
Avenida Professor Gama Pinto 2
1649-003 Lisboa
Portugal