This article is concerned with the existence and orbital stability of standing waves for a
nonlinear Schrödinger equation (NLS) with a nonautonomous nonlinearity.
It continues and concludes the series of
papers [6, 7, 8]. In , the authors make use of a continuation argument
to establish the existence in $R \times H^1$(RN$)$ of a smooth local branch of
solutions to the stationary elliptic problem associated with (NLS) and hence the existence
of standing wave solutions of (NLS) with small frequencies.
Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point
$(0,0)\in R \times H^1$(RN$)$ occur, or instability and asymptotic bifurcation occur.
The main hypotheses in  concern the behaviour of the nonlinearity with respect
to the space variable at infinity.
The paper  extends the results of  to (NLS) with more general nonlinearities.
In , the global continuation of the local branch obtained in  is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry
with respect to the space variable is assumed.
The aim of the present work is to prove the existence and discuss the orbital stability of
standing waves with high frequencies, independently of the results
obtained in  and . The main hypotheses now concern the
behaviour of the nonlinearity with respect to the space variable around the origin.
The methods are the same in spirit as that of  and permit to discuss the
asymptotic behaviour of the global branch of solutions obtained in .