Orbital stability of standing waves of a semiclassical nonlinear Schrödinger-Poisson equation

We study the orbital stability of single-spike semiclassical standing waves of a non-homogeneous in space nonlinear Schroedinger- Poisson equation. When the nonlinearity is subcritical or supercritical we prove that the nonlocal Poisson-term does not influence the stability of standing waves, whereas in the critical case it may create instability if its value at the concentration point of the spike is too large. The proofs are based on the study of the spectral properties of a linearized operator and on the analysis of a slope condition. Our main tools are perturbation methods and asymptotic expansion formulas.