We begin to study in this paper orbital and asymptotic stability of standing
waves for a model of Schr\"odinger equation with concentrated nonlinearity in
dimension three. The nonlinearity is obtained considering a {point} (or
contact) interaction with strength α, which consists of a singular
perturbation of the laplacian described by a selfadjoint operator Hα​,
where the strength α depends on the wavefunction: iu˙=Hα​u,
α=α(u). If q is the so-called charge of the domain element u,
i.e. the coefficient of its singular part, we let the strength α depend
on u according to the law α=−ν∣q∣σ, with ν>0. This
characterizes the model as a focusing NLS with concentrated nonlinearity of
power type. For such a model we prove the existence of standing waves of the
form u(t)=eiωtΦω​, which are orbitally stable in the
range σ∈(0,1), and orbitally unstable for σ≥1. Moreover,
we show that for σ∈(0,2​1​) every standing wave is
asymptotically stable in the following sense. Choosing initial data close to
the stationary state in the energy norm, and belonging to a natural weighted
Lp space which allows dispersive estimates, the following resolution holds:
u(t)=eiω∞​tΦω∞​​+Ut​∗ψ∞​+r∞​,ast→+∞, where U is the
free Schr\"odinger propagator, ω∞​>0 and ψ∞​,
r∞​∈L2(R3) with ∥r∞​∥L2​=O(t−5/4)ast→+∞. Notice that in the present model the
admitted nonlinearity for which asymptotic stability of solitons is proved is
subcritical.Comment: Comments and clarifications added; several misprints correcte