On Schrödinger-Poisson Systems

Abstract.

We discuss some recent results dealing with the existence of bound states of the nonlinear Schrödinger-Poisson system

$$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$

as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.