On the existence of ground states for nonlinear Schrödinger–Poisson equation

This paper is concerned with the existence of ground states for the Schrödinger–Poisson equation i∂tu=−∂x2u+V(u)u−f(|u|2)u, where V(u)V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities.