  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 $\alpha$, which consists of a singular
perturbation of the laplacian described by a selfadjoint operator $H_{\alpha}$,
where the strength $\alpha$ depends on the wavefunction: $i\dot u= H_\alpha u$,
$\alpha=\alpha(u)$. If $q$ is the so-called charge of the domain element $u$,
i.e. the coefficient of its singular part, we let the strength $\alpha$ depend
on $u$ according to the law $\alpha=-\nu|q|^\sigma$, with $\nu > 0$. 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 $u (t)=e^{i\omega t}\Phi_{\omega}$, which are orbitally stable in the
range $\sigma \in (0,1)$, and orbitally unstable for $\sigma \geq 1.$ Moreover,
we show that for $\sigma \in (0,\frac{1}{\sqrt 2})$ 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
$L^p$ space which allows dispersive estimates, the following resolution holds:
$u(t) = e^{i\omega_{\infty} t} \Phi_{\omega_{\infty}} +U_t*\psi_{\infty}
+r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty$, where $U$ is the
free Schr\"odinger propagator, $\omega_{\infty} > 0$ and $\psi_{\infty}$,
$r_{\infty} \in L^2(\R^3)$ with $\| r_{\infty} \|_{L^2} = O(t^{-5/4}) \quad
\textrm{as} \;\; t \rightarrow +\infty$. Notice that in the present model the
admitted nonlinearity for which asymptotic stability of solitons is proved is
subcritical.
