ORBITAL STABILITY OF STANDING WAVES OF A SEMICLASSICAL NONLINEAR SCHR ODINGER-POISSON EQUATION

We study the orbital stability of single-spike semiclassical standing waves of a nonhomogeneous in space nonlinear Schrodinger-Poisson equation. When the nonlinearity is subcritical or supercritical we prove that the nonlocal Poisson-term does not in uence 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.