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We begin to study in this paper orbital and asymptotic stability of standingwaves for a model of Schr\"odinger equation with concentrated nonlinearity indimension three. The nonlinearity is obtained considering a {point} (orcontact) interaction with strength $\alpha$, which consists of a singularperturbation 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 $\alpha$ dependon $u$ according to the law $\alpha=-\nu|q|^\sigma$, with $\nu > 0$. Thischaracterizes the model as a focusing NLS with concentrated nonlinearity ofpower type. For such a model we prove the existence of standing waves of theform $u (t)=e^{i\omega t}\Phi_{\omega}$, which are orbitally stable in therange $\sigma \in (0,1)$, and orbitally unstable for $\sigma \geq 1.$ Moreover,we show that for $\sigma \in (0,\frac{1}{\sqrt 2})$ every standing wave isasymptotically stable in the following sense. Choosing initial data close tothe 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 thefree 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 theadmitted nonlinearity for which asymptotic stability of solitons is proved issubcritical.

Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three